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PreCalculus

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Westchester Community CollegePreCalculus

Course Requirements

Professor Ted Cann Email: [email protected]

Office Hours: before class by appt.

TEXTBOOK:

CALCULATOR: TI-83, TI-83+, TI-84, or TI-84+ Graphing Calculator

ATTENDANCE: Attendance will be taken each class. Any student who is absent 2 times or less will have their lowest test grade dropped. This is not a right, it is a privilege. If you do not meet the attendance requirement, you will not be eligible to drop your lowest grade.*If you MUST be absent, I expect an email PRIOR to the beginning of class.*

CLASS PARTICIPATION: Please come to each class prepared with your note packets and graphing calculator. Preparedness is essential to participation. Please be prepared to participate in class at all times. Your grade will reflect your preparedness and participation.

LATENESS: Please arrive to class on time and be ready to learn. Lateness is a distraction to the class as a whole. Habitual lateness will have serious consequences. Please be aware that: 3 times late = 1 time absent (this could have a direct impact on your ability to drop your lowest test grade).

HOMEWORK ASSIGNMENTS: Homework will be assigned at every class session. Be prepared to hand in your work for a grade at any time. This may or may not be announced in advance.

TESTS: Tests will be scheduled with a reasonable amount of notice. Therefore, there will be no make-ups allowed except in extreme cases (family emergency, swine flu, etc.)

GRADING POLICY: Final Exam (cumulative) 20%Tests 50%Quizzes 20%Attendance/Participation 10%

COURSE WITHDRAWAL: Any student may withdraw from this course with a grade of W at any time until _______________. After then, a student who does not complete the course will receive a grade of F. The grades of WP, WF, and I are awarded only in exceptional circumstances beyond the student’s control. This is in full compliance with WCC policy.

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CHAPTERS

Chapter 2: Functions and Graphs2.1 Basics of Functions and Their Graphs2.2 More on Functions and Their Graphs2.5 Transformations of Functions

[*Emphasize the 6 basic functions, y=c , y=x , y=|x|, y=x2 , y=x , y=√x ]2.6 Combinations of Functions; Composite Functions2.7 Inverse Functions

Chapter 3: Polynomial and Rational Functions3.2 Polynomial Functions and Their Graphs3.3 Dividing Polynomials; Remainder and Factor Theorems3.4 Zeros of Polynomial Functions (*Omit Descartes’s Rule of Signs) 3.5 Rational Functions and Their Graphs3.7 Modeling Using Variation (optional topic)

Chapter 5: Trigonometric Functions5.5 Graphs of Sine and Cosine Functions 5.6 Graphs of Other Trigonometric Functions 5.7 Inverse Trigonometric Functions5.8 Applications of Trigonometric Functions

Chapter 6: Analytic Trigonometry6.1 Verifying Trigonometric Identities6.2 Sum and Difference Formulas6.3 Double-Angle, Power-Reducing, and Half Angle Formulas6.4 Product-to-Sum and Sum-to-Product Formulas6.5 Trigonometric Equations

Chapter 7: Additional Topics in Trigonometry7.1 Law of Sines7.2 Law of Cosines7.3 Polar Coordinates (optional topic)

Chapter 10: Conic Sections and Analytic Geometry10.1 Ellipse10.2 Hyperbola10.3 Parabola

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Precalculus Name:______________________________Review Material: Functions and Graphs

Date:_______________________________

Objective: To work in groups to rediscover the properties of functions and their graphs from Math 3.

LINEAR FUNCTIONS:

Forms of Linear Equations:

slope-intercept form: standard form:

point-slope form

PBLM SET1. Write the linear equation in slope intercept, standard, and point-slope form given the line passes through

(5, 2) and (7, 9)

2. Write the equation of the horizontal line that passes through (-9, 2)

Parallel & Perpendicular LinesParallel lines have __________ slopes.Perpendicular lines have slopes that are ____________ ______________.

3. Write the linear equation in standard form given that the line passes through (-2, 10) and is parallel to

the graph of y=−3 x− 4

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4. Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of y=2

3x+ 4

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Precalculus Name:______________________________Lesson: Linear Modeling

Date:_______________________________

Objectives: write linear functions to model a situation use linear functions to model and determine depreciation

DO NOW: The fixed costs per day for a ski manufacturer are $3,750, and the variable costs are $68 per pair of skis produced. If x pairs of skis are produced daily, express the daily cost C(x) as a function of x.

(1) A photo-copier was purchased by a law firm for $8,000 and is assumed to have a salvage value of $1,000 after 5 years. Its value has depreciated linearly during this period.

(a) Write a linear function D(t) that relates the value of the photo-copier in dollars to time t in years.

(b) What would be the depreciated value of the photo-copier after 1 year? 3 years?

(c) What is the practical domain of the depreciation function?

(d) Explain what the slope of this linear function represents.

(e) Sketch the graph and indicate the WINDOW used.

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(2) A clothing store marks up the price of a shirt from $20 to $33, and a jacket from $60 to $93. The markup policy of the store for items that cost over $10 is assumed to be linear.

(a) Write a linear function R(c) that expresses retail price R in terms of cost c.

(b) What does this store pay for a suit that retails for $240?

(c) What is the practical domain of this retail function?

(d) Explain what the slope of this linear function represents.


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Evaluating Functions: substitute numerical value or variable into function equation and simplify

(1) find f(-1) if f(x) = x2 – 1

____________________________________________________________________________________(2) find h(3) if h(x) = 3x2

(3) find f(-7) if f(w) = 16 + 3w – w2

(4) find g(m) if g(x) = 2x6 – 10x4 – x2 + 5

(5) find k(w + 2) if k(x) = 3x + 4

(6) find h(a – 2) if h(x) = 2x2 – x + 3

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Precalculus Name:__________________________________Lesson- Composition & Inverse of Functions

Date:___________________________________

Objective: To know how to find the composition and inverse of a function. To understand the process for finding the composition and inverse of a function. To be able to recognize an inverse graphically.

Do Now: Evaluate f ( x )=x3−x for x=2

Composition of Functions“following” one function with another.

Notation:Both of the following mean “f following g.”f (g ( x )) and ( f ∘g )(x )

g(f(x) and ( g∘ f )(x )

Ex 1:

f ( x )=x+5g( x )=4 x

Find: a) f (g ( x )) b) f (g (2) ) c) g( f ( x )) d) ( g∘ f )(3 )

Would you say that a composition is a commutative operation? Why/why not?

Ex 2:

h( x )=x2

r ( x )=x+3 Find: a) h(r (x )) b) r (h (x )) c) h(r (5 ))

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x

y

Inverse Functions

Steps:1. Write the equation in terms of x and y.2. Switch the x with the y.3. Solve for y.

PBLM SET

1. Find the inverse of y=4 x−8 2. Find the inverse of f ( x )=5 x−2

3. Graph each of the lines in examples one and two and their inverses on the same set of axes and identify any interesting characteristics.

4. Given the function f ( x )=− 1

3x−2

(a) Algebraically, find f -1(x).

(b) Algebraically, verify your answer to part (a).

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Precalculus Name:______________________________Lesson- Evaluate functions with variables;

difference quotient Date:_______________________________

Objectives: evaluate functions as expressions that involve one or more variables explore functions by evaluating and simplifying a difference quotient

Evaluating & Simplifying a Difference Quotient:

(1) For f(x) = x2 + 3x + 7, evaluate and simplify:

(a) f ( x+h ) (b)f ( x+h )− f ( x )

h, h≠0

(2) For f(x) = 3x2 – 2, evaluate and simplify:

(a) f ( x+h )

(b)f ( x+h )− f ( x )

h, h≠0

(c)f ( x )−f (a)

x−a, x−a≠0

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(3) For each given function, find and simplify:

f ( x+h )−f ( x )h

, h≠0

(a) f(x) = -x2 – 2x – 4

(b) f(x) = -2x + 5

(c) f(x) = x2 + 4x + 5

(4) For the function f(x) = 4x – 7, find and simplify:

(a)f ( x+h )−f ( x )

h, h≠0

(b)f ( x )−f (a)

x−a, x−a≠0

(5) For the function f(x) = -5x + 2, find and simplify:

(a)f ( x+h )−f ( x )

h, h≠0

(b)f ( x )−f (a)

x−a, x−a≠0

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Precalculus Name:______________________________Lesson- Symmetry, Odd/Even/Neither Functions

Date:_______________________________

Objectives: ~To learn an algebraic method for testing for symmetry with respect to axes and origin~To learn an algebraic method for determining whether functions are even/odd/neither~To learn shortcut approaches for the above

Odd Functions symmetric with respect to the origin TEST: f(-x) = -f(x)

Even Functions symmetric with respect to the y-axis TEST: f(x) = f(-x)

Symmetry Tests

symmetric with respect to the: the given equation is equivalent when:y-axis x is replaced with -xx-axis y is replaced with –yorigin x and y are replaced with –x and -y

Examples:

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RationalFunctions

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Precalculus Name:____________________________________Lesson- Finding zeroes, vertical and horizontal asymptotes

Date:_____________________________________

Objectives: Review of writing the domain of a graph in interval notation Determine the domain and x-intercepts for rational functions Determine any vertical or horizontal asymptotes

Interval Notation:Changing from Interval to Inequality Notation. (and vice versa)

Complete the following table:

Interval Notation Inequality Notation[a,b]

a≤x<b(a,b]

a< x<b[ b ,∞)

x>b(−∞ , a)

x<a

Rational Functions:

Vertical Asymptote:

Horizontal Asymptote: Three cases for any rational function: f ( x )=

am xm+. . .+a1 x+a0

bn xn+. ..+b1 x+b0

Case: m<n m=n m>n

Horizontal Asymptote y=0 y=

am

annone

Zeros of a Function:

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Precalculus Name:____________________________________Lesson- Graphing Rational Functions

Date:_____________________________________

Objectives: Graphing Rational Functions without the graphing calculator by:a) determining zeroes of the rational functionb) determining vertical and horizontal asymptotesc) determining slant asymptotesd) describing the behavior of the function around vertical asymptotes

Do Now: Given the function f(x) = x3 – 6x2 + 10x – 8

(1) What is the degree of this polynomial?

(2) What is the leading coefficient?

(3) Determine if 4 is a zero of this function f(x):

Graph each (on separate graph paper) of the following functions (and check with the graphing calculator) by:

a) determining zeroes of the rational functionb) determining vertical and horizontal asymptotesc) determining slant asymptotesd) describing the behavior of the function around vertical asymptotes

(increasing/decreasing)

(1) f ( x )= x

x−1

(2) f ( x )= 5x

x2+4

(3) f ( x )=2 x2+3 x+7

x2−4

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Precalculus Name:____________________________________Lesson- Jump Discontinuities

Date:_____________________________________

Objective: To learn the process for graphing a rational function that includes a jump discontinuity (hole)

DO NOW: Sketch the graph of the following rational function without the use of a calculator.

g( x )= 5−xx2−4

__________________________________________________________________________________________Jump Discontinuity

Process

Examples

Sketch the graph of the following rational functions.

1.f ( x )= x−2

x2−4 2.g( x )= x2+4 x+4

x+2 3.h( x )= 2−x

x2−5 x+6

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Precalculus Name:____________________________________Lesson- Oblique Asymptotes

Date:_____________________________________

Objectives: Determine existence of any oblique (slant) asymptotes

Oblique Asymptotes (aka Slant Asymptotes):

Process: Divide the numerator by the denominator using long division Drop any remainder The linear function that is the result is the slant asymptote

**note: there is only a slant asymptote if the degree of the numerator is one more than that of the denominator.

Determine the oblique asymptote for each rational function and graph each:

1. f ( x )= x2−x

x+1 2. f ( x )= x3+1

x23.

f ( x )= x2+x−6x+1

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x

y

Precalculus Name:____________________________________Lesson- More Rational Function Graphs

Date:_____________________________________

Objective: To practice with graphing rational functions with discontinuities and radicals.__________________________________________________________________________________________

Do Now: Find the domain in interval notation and any asymptotes for h( x )= 3 x

x+5

__________________________________________________________________________________________What is a discontinuity?

Horizontal Asymptotes for rational functions with a radical in the numerator:

Degree Relationship m<n m=n m>n

Horizontal Asymptote y=0 y=±am

an

none

In small groups, graph each of the following on the grids provided:

1. f ( x )= x2+6 x+8

x2−x−2

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x

y

x

y

2. f ( x )= x2−4

x−2

3. f ( x )=3√x2+1

x−1

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Precalculus Name:____________________________________WKST- More practice with rational functions and graphs

Date:_____________________________________

Objectives: more practice with determine vertical, horizontal, and oblique asymptotes more practice with graphing rational functions

Showing all work, find all vertical, horizontal, and oblique asymptotes. Do not graph.

(1)f ( x )= 2 x2

x−1 (2)h( x )= x2−x

x+1

(3)q ( x )= 2 x

x2+2 (4)r ( x )=2 x2−3 x+5

x

(5)c ( x )=5 x2−10 x+1

x−2 (6)k (x )= 3 x2+7

9x2−36

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x

y

x

y

Showing all work, graph each function:

(7)g( x )= 1

x−6

(8)t ( x )= 3 x

5 x+5

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x

y

x

y

Showing all work, graph each function:

(9)p( x )= 4

x2−4

(10)s( x )= x

x2−1

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WALL

x

x

Precalculus Name:__________________________________Lesson: Mathematical Modeling w/

Rational Functions Date:___________________________________

Objectives: write polynomial functions to model a situation determine possible dimensions within a practical domain

A rectangular area of 60 square meters has a wall on one of its sides, as shown. The sides perpendicular to the wall are made of fencing that costs $36 per meter. The side parallel to the wall is made of decorative fencing that costs $48 per meter.

(a) Write a function to express the total cost C(x) of the fencing as a function of x.

(b) Find the minimum cost to the nearest dollar. What are the dimensions that give this minimum cost to the nearest tenth?

(c) Sketch the graph and indicate the WINDOW used.

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W A L L

W

L

x x x

4 4 4

(3) Carmelo decides to build a dog kennel with 80 feet of fencing wire against the side of his house, such that one of the lengths of the fence will be replaced by the wall.

(a) Write a perimeter (total fencing) equation.

(b) Write a function to express the total area A(W) in terms of W.

(c) What is the practical domain?

(d) Find the dimensions of W and L, to the nearest tenth, such that the diagram above would have a maximum area. What is the maximum area to the nearest tenth?


(4) An animal clinic wants to construct a kennel with 3 individual pens, each with a gate 4 feet wide and an area of 90 square feet. The fencing does not include the gates.

(a) Write a function to express the fencing F(x) as a function of x.

(b) Find the dimensions for each pen, to the nearest tenth of a foot that would produce the required area of 90 square feet but use the least fencing. What is the minimum fencing to the nearest tenth?

(c) Sketch the graph and indicate the WINDOW used.

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PolynomialFunctions

Coming into this unit, you should be proficient in the following topics: Factoring Solving quadratics in a variety of ways (esp. completing the square) Writing quadratics in standard form Knowing the difference between a factor and a root Complex Numbers

Precalculus Name:____________________________________Lesson- Synthetic Division and the Remainder Theorem

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Date:_____________________________________

Objective: To use the synthetic division process as well as the remainder theorem to: Divide polynomials more quickly than the long division process Evaluate a function given the independent variable

Do Now: Find the roots of the following quadratic:2 x2−6 x+3=0

Synthetic Division List out the coefficients of the dividend in descending order (put a 0 in for any missing powers) Write the divisor in x−r form and bring down the first coefficient of the dividend Multiply r by the 1st term and add to the second term Multiply r by the result of the previous step and add to the next term, repeat if necessary The last result is a remainder, treat it the same way you would in long division

Remainder Theoremf (r ) is the remainder (coefficient) after the synthetic division process.

Examples:

1. Divide y2+3 y+2 by y+2using synthetic division. 2.

3 x 4−11 x3−18 x+8x−4

3. Find f (−3 ) if f ( x )=4 x4+10 x3+19 x+5 using the remainder theorem.

Precalculus Name:____________________________________

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Lesson: Fundamental Theorem of algebra &The Rational Root Theorem & The Date:_____________________________________Imaginary Root Theorem

Objective: to determine roots of polynomial equations by applying the Fundamental Theorem of Algebra to be able to use the rational root theorem to find the factors and roots of any polynomial to be able to use the imaginary root theorem to find complex roots of any polynomial

Fundamental Theorem of Algebra: In mathematics, the fundamental theorem of algebra states that every complex polynomial p(z) in one variable and of degree n ≥ 1 has some complex root.

Complex Numbers:

Examples:

Write the polynomial equation using the given roots:(1) 9 and 7 (2) -6 and 4 (3) -3 and -5

Write P(x) as a product of first-degree factors using the given zero:

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http://en.wikipedia.org/wiki/Root_(mathematics)

http://en.wikipedia.org/wiki/Degree_of_a_polynomial

http://en.wikipedia.org/wiki/Polynomial

http://en.wikipedia.org/wiki/Fundamental_theorem

http://en.wikipedia.org/wiki/Mathematics

(1) P(x) = x3 + 9x2 + 24x + 16; -1 is a zero

(2) P(x) = 2x3 + 7x2 – 19x – 60; 3 is a zero

(3) P(x) = x3 – 4x2 – 3x + 18; 3 is a double zero

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Rational Roots Theorem : Possible roots = p

q , where p represents factors of the constant term and q represents factors of the leading coefficient.

Imaginary Roots Theorem: Imaginary roots occur in complex conjugate pairs.

Process:1. Use Rational Roots Theorem to find potential rational roots.

2. Use synthetic division, or long division, to find an actual root.

3. Repeat step 2 until the polynomial is of degree 2.

4. Factor the remaining quadratic polynomial (it is possible to get imaginary answers).

5. List all factors for “completely factored form”.

More examples:

1. 2.

3. 4.

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5. 6.

7. 8. x3−2 x2+4 x−8

9.

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Write the polynomial equation using the given roots:

(10) 4, 6i, and -6i (11) 2, -5i, and 5i

Write P(x) as a product of first-degree factors using the given zero:

(12) P(x) = x3 – 16x2 + 86x – 156; 6 is a zero

(13) P(x) = x4 – 6x3 + 5x2 + 32x + 20; -1 is a double root.

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x x

x

x x

x

x

x

20 cm

30 cm

Precalculus Name:__________________________________Lesson- Polynomial Function Modeling

Date:___________________________________

Objectives: write polynomial functions to model a situation determine possible dimensions within a practical domain determine the area or volume

Def.: Practical Domain:

(5) A box is to be made out of a piece of sheet metal that measures 20 cm by 30 cm. Squares, x cm on a side, will be cut from each corner, and then the ends and sides will be folded up to create a box that will have no top.

(a) Write a function to express the volume of the box V(x) in terms of x.

(b) What is the practical domain?

(c) Find the dimensions of the box, to the nearest tenth, that would give the maximum volume. What is the maximum volume to the nearest tenth?

(d) Sketch the graph and indicate the WINDOW used.

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TRIGONOMETRY UNIT NOTESPART 1

Coming into this unit, you should be proficient in the following topics: SOHCAHTOA The Unit Circle Finding EXACT values of trig functions Radians v. Degrees

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De 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360

Ca. 510 a.d. Ca. 1114-1187 a.d. 1551 1583 1585-95 1620 1658 1674 1700

Edmund Gunter coins the term: cotangens, which later becomes cotangent or “complement of tangent”

The first printed table of secants appeared in the work Canon doctrinae triangulorumBy Leipzig

ardha-jya half chord is turned around to jya-ardha (“chord-half”) which in due time is shortened to jya or jiva (or bowstring)

Trigonometric Function Origin TimelineHindu text Aryabhatiya of Aryabhata is written and refers to a half-chord known as ardha-jya

Sir Jonas Moore introduces “Cos” as the abbreviation for “cosinus”

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g

Rad 0 π6

1 π6

1 π4

2 π6

3 π6

, 2 π4

4 π6

3 π4

5 π6

6 π6 ,

4 π4

7π6

5 π4

8 π6

9 π6 ,

6 π4

10 π6

7 π4

11 π6

12 π6 ,

8 π4

Sin √02

√12

√32

√12

√02

−√12

−√22

−√32

−√42

−√32

−√22

−√12

√02

Cos √12

√02

−√12

−√22

−√32

−√42

−√32

−√22

−√12

√02

√12

√32

Tan √1√3

√2√2

√3√1

√4√0

−√3√1

−√2√2

−√1√3

−√0√4

√1√3

√2√2

√3√1

−√4√0

−√3√1

−√2√2

−√1√3

Csc 2√0

2√1

2√2

2√3

2√4

2√3

2√2

2√1

2√0

− 2√1

− 2√2

− 2√3

− 2√4

− 2√3

− 2√2

− 2√1

2√0

Sec 2√4

2√3

2√2

2√1

2√0

− 2√1

− 2√2

− 2√3

− 2√4

− 2√3

− 2√2

− 2√1

2√0

2√1

2√2

2√3

2√4

Cot √4√0

√3√1

√2√2

√1√3

−√1√3

−√2√2

−√3√1

−√4√0

√3√1

√2√2

√1√3

−√0√4

−√1√3

−√2√2

−√3√1

√4√0

22

22

22

22

40

40

40

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Precalculus Name:____________________________________Lesson: Review of Unit Circle Processes

Date:_____________________________________

Objective: Finding sine and cosine EXACTLY

DO NOW: Determine the exact value of sin (-45) without using the calculator.

Sketch a figure and find the coordinates for each circular point:

1.( 8π

3 )2.

(−5 π6 )

3.(−7 π

6 )4.

(11 π3 )

Find the sine, cosine, and tangent of each radian measure:

5.

π2 6.

3π4 7. 2 π

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Precalculus Name:____________________________________Lesson: Radians and Arc Length

Date:_____________________________________

Objective: Discuss the relationship among central angles, radii and arc lengths.

DO NOW: Determine the exact value of tan (-45) without using the calculator.

__________________________________________________________________________________________What is a central angle? What is an arc length?

Relationship among central angle, radius and arc length:

Examples:1. Find the measure of a positive central angle that intercepts an arc of 14 cm on a circle of radius 5 cm.

2. Find the length of the arc intercepted by a central angle of 3.5 radians on a circle of radius 6 m.

3. A wheel of radius 18 cm is rotating at a rate of 90 revolutions per minute.a. How many radians per minute is this?b. How many radians per second is this?c. How far does a point on the rim of the wheel travel in one second?d. Find the speed of a point on the rim of the wheel in centimeters per second.

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Precalculus Name:_______________________________Lesson: Working with central angles, radii and Arc lengths Date:________________________________

Objective: find the length of an arc given the measure of the central angle find the radian measure of a central angle given an arc and the radius

(1) Find the measure of a central angle opposite an arc of 24 cm in a circle with a radius of 4 cm.

(2) Given a central angle of 2 π3 , find the length of its intercepted arc in a circle of radius 14 cm, rounding to

the nearest tenth.

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(3) Given a central angle of 125, find the length of its intercepted arc in a circle of radius 7 feet, rounding to the nearest tenth.

(4) An arc is 14.2 cm long and is intercepted by a central angle of 60. Rounding to the nearest tenth, what is the radius of the circle?

(5) The diameter of a circle is 22 inches. If a central angle measures 78, find the length of the intercepted arc to the nearest tenth.

(6) If the pendulum of a grandfather clock is 44 in long and swings through an arc of 6, find the length, to the nearest tenth of an inch, of the arc that the pendulum traces.

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Precalculus Name:_____________________________HW- Working with radian measure

Date:______________________________

SHOW ALL WORK:

(1) Find the exact radian measure of a central angle opposite an arc of 30 feet in a circle of radius 12 feet.

(2) Given a central angle of 128, find the length of its intercepted arc in a circle of radius 5 centimeters. (Round to the nearest tenth.)

(3) An arc is 12 yards long and is intercepted by a central angle of 3 π11 radians. Find the radius of the circle.

(Round to the nearest tenth.)

(4) Find the exact radian measure of a central angle opposite an arc of 27 meters in a circle of radius 18 meters.

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A

B

C D

E

1.8 mi

1.46 mi0.67 mi

0.70 mi

84.580

SHOW ALL WORK:

(5) Given a central angle of 147, find the length of its intercepted arc in a circle of radius 10 meters. (Round to the nearest tenth.)

(6) An arc is 70.7 meters long and is intercepted by a central angle of 5 π4 radians. Find the diameter of the

circle. (Round to the nearest tenth.)

(7) The figure below shows a stretch of roadway where the curves are arcs of circles. Find the length of the road from point A to point E. (Round to the nearest hundredth.)

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Precalculus Name:____________________________________Lesson: Area of sectors and arc lengths of a circle

Date:_____________________________________

Objective: find the area of a sector find the area of a segment

Sector of a Circle:

(1) Find the area of a sector if the central angle measures 3 π7 radians and the radius of the circle is 11 cm.

(Round to the nearest tenth.)

(2) Find the area of a sector if the central angle measures 315 degrees and the diameter of the circle is 8 feet. (Round to the nearest tenth.)

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Segment of a Circle:

(3) Find the segment of a circle with a radius of 6 feet and a central angle of 108 degrees. (Round to the nearest tenth.)

(4) Find the area of the shaded region if a pentagon is inscribed in a circle that has a radius of 7 inches. (Round to the nearest tenth.)

(5) A windshield wiper on a car is 20 in long and has a blade 16 in long. If the wiper sweeps through an angle of 110, how large an area does the wiper blade clean to the nearest square inch?

44

Precalculus Name:____________________________________Lesson- Geometric Applications of Trig

Date:_____________________________________

Objective: solve geometric applications using trigonometric functions

ANSWER THE FOLLOWING QUESTIONS SHOWING ALL WORK AND ROUND TO THE NEAREST TENTH:

(1) An octagon is inscribed within a circle that has a diameter of 15 centimeters. Find the area of shaded region (which is eight segments).

(2) A sector has an area of 15.3 square meters. The radius of the circle is 3 meters.(a) Find the radian measure of the central angle.(b) Find the degree measure of the central angle.(c) Find the arc length of the sector.

(3) Find the area of a regular pentagon that is inscribed in a circle with a diameter of 7.3 feet.

45


(4) The adjacent sides of a parallelogram measure 14 centimeters and 20 centimeters, and one angle measures 57. Find the area of the parallelogram.

(5) The base of an isosceles triangle is 48.8 ft long and its vertex angle measures 38.6. Find the length of each leg.

(6) A small rectangular park is crossed by two diagonal paths, each 280 m long, that intersect at a 34 angle. Find the dimensions of the park.

Precalculus Name:_______________________________46

Lesson- More geometric applications of trigDate:________________________________

Objective: solve geometric applications using trigonometric functions


(1) If a regular pentagon is inscribed in a circle of radius 5.35 centimeters, find the length of one side of the pentagon.

(2) If a circle of radius 4 feet has a chord of length 3 feet, find the central angle that is opposite this chord to the nearest degree.

47


(3) Find the perimeter of a square inscribed in a circle of radius 5 centimeters.

(4) The sides of a parallelogram are 20 centimeters and 32 centimeters long. If the longer diagonal measures 40 centimeters, find the measures of the angles of the parallelogram.

(5) Each base angle of an isosceles triangle measures 4230. The base is 14.6 meters long.(a) Find the length of a leg of the triangle.(b) Find the altitude of the triangle.(c) Find the area of the triangle.

48

Precalculus Name:____________________________________Lesson- Inverse Trig Functions

Date:_____________________________________

Objective: To learn to use inverse trig functions to solve for an angle or angles

DO NOW: Find the exact value of sec 5 π

4

__________________________________________________________________________________________What is an inverse trig function? What is it used for?

Examples:

1. Write in the form of an inverse function: cos β=δ

2. Write in the form of an inverse function:cos 45 °= √2

2

3. Solve by finding the value of x to the nearest degree: Sin−1 (−1 )=x

4. Solve by finding the value of x to the nearest degree: Arc cos 1

2=x

Find each value (put angles in radian measure). Round any decimals to the nearest hundredth.

5.Arc tan(−√3

3 )6.

cos (2 Sin−1 √32 )

49

Precalculus Name:____________________________________Lesson: Law of Cosines

Date:_____________________________________

Objective: solve triangles by using the Law of Cosines

Law of Cosines:

1. Suppose a triangle ABC has side a = 4, side b = 7, and angle C = 54º. What is the measure of side C?

2. Suppose a triangle XYZ has sides of x = 5, y = 6, and z = 7. What is the measure of the angle across from the side of measure 6?

3. Suppose a triangle ABC has side b = 2, side a = 5, and angle B = 27º. Find the measure of side c.

4. Suppose a triangle ABC has side b = 4, side a = 5, and angle B = 27º. Find the measure of side c.

50

Precalculus Name:____________________________________Lesson- Forces and the Law of Cosines

Date:_____________________________________

Objective: To determine the resultant force vector when given two force vectors and an included angle.

DO NOW: If m∠ A=30 ° , AC=5, and AB=7, solve the triangle. Find all sides to the nearest tenth and angles to the nearest degree.

__________________________________________________________________________________________Force- push or pull upon an object resulting from the object's interaction with another object.

Vector- a quantity of force having both magnitude and direction.

Examples:

1. Two forces separated by 52 degrees acts on an object at rest. The magnitude of the two forces are 32 Newtons and 17 Newtons. Find the resultant force vector to the nearest Newton.

51

2. A game of “Three Way Tug-O-War” is being played by a group of students. Two of the students are trying to gang up on the other. They believe that it will be easier to win if they increase the angle they create with the third person. Is that true? Justify your answer by providing examples.

3. Two fisherman have hooked the same fish and they are trying to cooperatively reel it in. The angle the fisherman make with the fish is 87 degrees. If the first fisherman’s line has a maximum tensile strength 223 Newtons and the second fisherman’s line has a maximum tensile strength of 401 Newtons and the fishermans’ lines are at maximum strain, what is the resultant force applied to the fish?

4. What is the angle separating two component force vectors whose magnitude are 15N and 17N respectively if the resultant vector is 21N?

52

Precalculus Name:____________________________________Lesson: Law of Sines, area of a triangle

Date:_____________________________________

Objective: solve triangles by using the Law of Sines find the area of a triangle

Law of Sines:

Area of Triangles:

(1) Given DEF where D = 29, E = 112, and d = 22:

(a) Solve DEF, rounding answers to the nearest tenth

(b) Find the area of DEF to the nearest tenth

53

(2) Given ABC where A = 13, B = 6520, and a = 35:

(a) Solve ABC such that:(i) C is in DMS form(ii) b is rounded to the nearest tenth(iii) c is rounded to the nearest tenth

(b) Find the area of ABC, to the nearest tenth, using the formula K= 1

2bc sin A

(3) Given GHJ where g = 45.7, H = 111.1, and J = 27.3:

(a) Solve GHJ, rounding answers to the nearest tenth

(b) Find the area of GHJ (to the nearest tenth)

Precalculus Name:____________________________________54

45

2

25 ft

122

40 m

25 m

110

23

120 m

WKST- Law of Sines/Cosines WPDate:_____________________________________

1. A lamppost tilts toward the sun at a 2 angle from the vertical and casts a 25 foot shadow. The angle from the tip of the shadow to the top of the lamppost is 45. Find the length of the lamppost to the nearest tenth of a foot.

2. A derrick at the edge of a dock has an arm 25 meters long that makes a 122 angle with the floor of the dock. The arm is to be braced with a cable 40 meters long from the end of the arm back to the dock. To the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened?

3. Using the picture seen to the right, and rounding to the nearest tenth of a meter, find the height of the tree.


55

Lesson: Law of Sines- The Ambiguous CaseDate:_____________________________________

Objective: solve triangles by using the Law of Sines (ambiguous case) find the area of a triangle

DO NOW: The sides of a triangle measure 6, 7, and 9. What is the measure of the smallest angle in the triangle?

Law of Sines:

__________________________________________________________________________________________Ambiguous Case: This is the case in the Law of Sines ( SSA) where there may be none, one, or two distinct

triangles for which you can solve.

Showing all work, find all solutions for each ABC. If no solutions exist, write none.Round all answers to the nearest tenth.

1. A = 42, a = 22, b = 12 2. b = 50, a = 33, A = 132

3. a = 125, A = 25, b = 150 4. a = 32, c = 20, A = 112

5. b = 15, c = 13, C = 50 6. a = 12, b = 15, A = 55

56



57

y

x

1

-1

Lesson- Graphing sine and cosine functionsDate:_____________________________________

Objectives: To construct graphs of the sine and cosine functions

Definitions:

Amplitude:

Frequency:

Period:

Graphing Trigonometric Functions:

(1) Graph y=sin x in the interval -2 x 2

Period: x-intercepts:

Domain: y-intercepts:

Range: Maximum point:

Minimum point:

58

y

x

1

-1

y

x

y

x

(2) Graph y=cos x in the interval -2 x 2

Period: x-intercepts:

Domain: y-intercepts:

Range: Maximum point:

Minimum point:

(3) Graph y=2sin 2 x in the interval - x 2

(4) Graph y=3cos ( 1

2x)

in the interval −π≤x≤3 π

2

59

y

x

y

x

y

x

Precalculus Name:______________________________HW- Sine and cosine graphs

Date:_______________________________

Objective: find the amplitude and period to graph sine and cosine functions

(1) y=−2sin x (−2π≤x≤2 π )

(2) y=4 cos x

3 (−3 π≤x≤3 π )

(3) y=1

2cos 2 x (−π≤x≤2 π )

60

(4)

61

y

x

y

x

2

y

x

y=2sin x4 (−2 π≤x≤2 π )

(5) y=1 .5 cos4 x (−π≤x≤π )

(6) y=−3 sin x

2 (−2π≤x≤2 π )

62

y

x

1

-1

y

x

2

y

x

(7) y=−sin x (−π≤x≤3 π2 )

(8) y=2cos ( 4 x ) (−π≤x≤π )

(9) y=1

2sin ( 1

2x ) (−4 π≤x≤4 π )

63

y

x

y

x

Precalculus Name:____________________________________Lesson: Phase shifts and translations

Date:_____________________________________

Objectives: find the phase shift for sine and cosine functions graph translations of sine and cosine functions

Phase Shift:

Translation:

(1) Graph y=sin ( x+π ) in the interval 0 x 4

(2) Graph y=cos (2 x−π

2 ) in the interval 0 x 2

64

y

x

y

x

y

x

(3) Graph y=3 cos (x+ π

2 ) in the interval -2 x 2

(4) Graph y=2sin ( x

2−π

8 ) in the interval -2 x 2

(5) Graph y=cos (2 x+π ) in the interval -2 x 2

66

y

x

y

x

y

x

(6) Graph y=2cos x−4 in the interval 0 x 4

(7) Graph y=2sin 2 x−2 in the interval -2 x 2

(8) Graph y=4 cos ( x

2+π )−6

in the interval 0 x 4

67

y

x

y

x

y

x

Precalculus Name:______________________________HW- Phase shift and translations

Date:_______________________________

GRAPH THE FOLLOWING TRIGONOMETRIC FUNCTIONS WITHIN THE GIVEN INTERVAL:

(1) y=cos ( x−π )+1 (−π≤x≤3 π )

(2) y=sin x

2+ 1

2 (−2π≤x≤3 π )

(3) y=2sin (x+ π

2 )−2(−2 π≤x≤2 π )

68

y

x

y

x

(4) y= 1

2cos ( 4 x−π ) (−π≤x≤π )

(5) y=3 sin ( x+ π

4 ) (−2 π≤x≤2 π )

69

y

x

y

x

Precalculus Name:____________________________________WKST- More practice with phase shift and translations

Date:_____________________________________

Objectives: find the phase shift and vertical translation for sine and cosine functions graph translations of sine and cosine functions

(1) Graph y=2cos x+1 in the interval 0 x 4

(2) Graph y=2sin x−2 in the interval -2 x 2

70

y

x

y

x


2−π )−6

in the interval 0 x 4


2 )−1 in the interval - x 2

71

Precalculus Name:_______________________________Lesson- Writing equations of sine and cosine functions

Date:________________________________

Objectives: write the equations of sine and cosine functions given the amplitude, period, phase shift, and vertical translation

Writing Trigonometric Functions:

Write an equation of the sine function with each given amplitude and period:

(1) amplitude = 4, period = 2

(2) amplitude = 35.7, period = π4

(3) amplitude = 0.8, period = 10

Write an equation of the cosine function with each given amplitude and period:

(4) amplitude = 58 , period =

π7

(5) amplitude = 0.5, period = 0.3


72

Write a sine function with each given period, phase shift, and vertical translation:

(7) period = 2, phase shift = 0, vertical translation = -6

(8) period = π2 , phase shift =

π8 , vertical translation = 0

(9) period = , phase shift = − π

4 , vertical translation = 3

Write a cosine function with each given period, phase shift, and vertical translation:

(10) period = 3, phase shift = , vertical translation = -1

(11) period = 5, phase shift = -, vertical translation = -6

(12) period = π3 , phase shift =

− π2 , vertical translation = 10

State the amplitude, period, phase shift, and vertical translation for each function:

(13)y=−7 .5 cos x

3 A = P = PS = VT =

(14)y= 1

4sin 6 x−8

A = P = PS = VT =

(15)y=−3

5sin (2 x+ π

4 )+9 A = P = PS = VT =

(16)y=cos (3 x−π

2 ) A = P = PS = VT =

73

Precalculus Name:______________________________HW- Writing the equation of sine and cosine functions

Date:_______________________________

Write an equation of the sine function with each given amplitude and period:


(2) amplitude = 0.5, period =

Write an equation of the cosine function with each given amplitude and period:

(3) amplitude = 37 , period = 0.2

(4) amplitude = 15 , period =

2 π5

Write a sine function with each given period, phase shift, and vertical translation:

(5) period = 2, phase shift = − π

2 , vertical translation = 12

(6) period = 8, phase shift = -, vertical translation = -2

Write a cosine function with each given period, phase shift, and vertical translation:

(7) period = , phase shift = − π

4 , vertical translation = -1

(8) period = 4, phase shift = π8 , vertical translation = 5

74

SHOW ALL WORK:

State the amplitude, period, phase shift, and vertical translation for each function:

(9) y=−2cos 0.5 x+3 A = P = PS = VT =

(10)y= 2

3cos 3 π

7x

A = P = PS = VT =

(11)y=3 sin (2x−π

2 )A = P = PS = VT =

(12)y=−1

3sin( x

3+ π

6 )−6A = P = PS = VT =

(13)y=−4 sin (4 x+ π

4 )−4A = P = PS = VT =

(14)y=8 sin( x

2− π

8 )+1A = P = PS = VT =

75


77

Precalculus Name:_____________________________HW- Rational Expressions

Date:_______________________________

Simplify each of the following and put all answers in simplest factored form.

1)

5 xx2+5x+6

− 3 xx2−4 2)

x2−x−302 x2−11 x−6

÷ 2 x2−11 x+124 x2−4 x−3

3)

3a+b

⋅a2−b2

2 a−b−5

a 4)

xx+1

+ x2

x2−13x

x−1− 2 x2

x+1

78

Trigonometry Unit: Formulas & Identities

Pythagorean and Quotient Identitiessin2 A + cos2 A = 1tan2 A + 1 = sec2 Acot2 A + 1 = csc2 A

tan A=sin Acos A

cot A=cos Asin A

Functions of the Sum of Two Anglessin (A + B) = sin A cos B + cos A sin Bcos (A + B) = cos A cos B – sin A sin B

tan( A+B)= tan A+ tan B1− tan A tan B

Functions of the Difference of Two Anglessin (A – B) = sin A cos B – cos A sin Bcos (A – B) = cos A cos B + sin A sin B

tan( A−B )= tan A− tan B1+ tan A tan B

Functions of the Double Anglesin 2A = 2 sin A cos A

cos 2A = cos2 A – sin2 Acos 2A = 2 cos2 A – 1cos 2A = 1 – 2 sin2 A

tan2 A= 2 tan A1−tan2 A

Functions of the Half Angle

sin 12

A=±√ 1−cos A2

cos 12

A=±√ 1+cos A2

79

tan 12

A=±√ 1−cos A1+cos A

Precalculus Name:______________________________Lesson- Proving Trig Identities

Date:_______________________________

Objective: To prove the validity of pythagorean, reciprocal and quotient identities.

DO NOW:

5 xx2−5 x+6

+ 3 xx2−4

____________________________________________________________________________________Pythagorean Identities

Identity: an equation that is true for all values of the variable

Ex: a2−b2=(a+b)( a−b ) or( ab )

2

=a2

b2

Pythagorean Identities

1st : cos2θ+sin2 θ=1This identity is from the equation of a circle centered at (0,0) and whose radius is 1.

From the previous identity, others can be obtained.

2nd : 1+ tan2 θ=sec2θ

3rd: cot2θ+1=csc2θ

Proving Trig IdentitiesThe idea is to show both sides of the equation can be written in the same form.

Helpful hints:1. Always start with the most complicated side

80

2. Look for algebraic identity that can be applied (ex:tanθ=sin θ

cosθ orcos2 θ+sin2 θ=1 )3. Try writing the expression in terms of sine and/or cosine.4. If you get stuck on one side, try the other!5. Never cross over the equal sign. Work with each side independently.

*To prove that an equation is an identity, you must show that it is true for all values of the variable for each side of the equation.Examples:

1. Prove:

1sin2θ

+ 1cos2θ

= 1sin2θ cos2 θ

2. Prove: sin4 θ−cos4θ=sin2 θ−cos2θ

3. Prove: tanθ+cot θ=csc θ secθ

81

4. Prove: csc x+cot xtan x+sin x

=cot xcsc x

Precalculus Name:______________________________CW/HW: Trig Identity Mixed Problem Set

Date:_______________________________

Objective: To use algebraic/proportion techniques in conjunction with Pythagorean, reciprocal, sum, difference, double angle and half angle rules to verify identities.

Verify each identity:

1. sec4 x−2sec2 x tan2 x+ tan4 x=1 2. (1−cos x )(csc x+cot x )=sin x

3.

1+ tan y1+cot y

=sec ycsc y 4.

cos2 x+cot xcos2 x−cot x

=cos2 x tan x+1cos2 x tan x−1

5. cos2 x (1−sec2 x )=−sin2 x 6.

tan y+cot ycsc y

=sec y

82

7.1+cos2 xsin 2 x

=cot x8.

11+sin x

+ 11−sin x

=2 sec2 x

Precalculus Name:______________________________HW: Trig Identity Mixed Problem Set

Date:_______________________________

Objective: To use algebraic/proportion techniques in conjunction with Pythagorean, reciprocal, sum, difference, double angle and half angle rules to verify identities.

HW Verify each of the following neatly on separate paper. Be sure to show all steps for full credit. WILL BE COLLECTED AND GRADED! [30 point quiz]

83

1 . cos x tan x=sin x2 . cot x cos x+sin x=csc x

3 . 1+sin xcos x

+cos x1+sin x

=2 sec x 4 . sin2 x+2sin x+1cos2 x

=1+sin x1−sin x

5 . tan x−cot xtan x+cot x =1−2 cos2 x6 . 3 cos2 z+5 sin z−5

cos2 z=

3 sin z−21+sin z

Precalculus Name:______________________________CW/HW- Proving Trig Identities #1

Date:______________________________

Verify each of the following identities:

1. sin xcot x=cos x 2. (cos x−sin x )2=1−2 sin xcos x

84

3. cos x ( tan x+sin x cot x )=sin x+cos2 x 4. cot x cos x+sin x=csc x

5.

(1−cos x ) (1+cos x )cos2 x

=tan2 x6. csc x−cos x cot x=sin x

7. tan2 x−sin2 x=tan2 x sin2 x 8.sec4 x−2sec2 x tan2 x+ tan4 x=1

9.

sin2 x+2sin x+1cos2 x

= 1+sin x1−sin x 10.

tan x+sec x=cos x1−sin x

11.

tan xsin x−2 tan x

= 1cos x−2 12.

1+sin xcos x

+cos x1+sin x

=2 sec x

Precalculus Name:______________________________Lesson: Sum, difference and ½ angle identities

Date:______________________________

85

Objective: use the sum, difference, and half-angle identities to evaluate trigonometric expressions

DO NOW: Prove:

cos2 y1−sin y

=1+sin y

Sum and Difference Identities

Half Angle Identities

Double Angle Identities

Precalculus Name:______________________________CW/HW- Trig Identities #2- Sum and Difference

Date:______________________________

86


1. sin( x− y )+sin( x+ y )=2sin x cos y 2. cos ( x− y )+cos ( x+ y )=2 cos x cos y

3.tan( x+ y ) tan( x− y )= tan2 x−tan2 y

1−tan2 x tan2 y 4.

cos ( x− y )sin xcos y

=cot x+ tan y

1.tan x− tan y=sin (x− y )

cos x cos y 6.

sin( x+ y )sin x sin y

=cot x+cot y

7.

cos ( x+ y )cos x cos y

=1−tan x tan y8.

tan( x− y )=sin( x− y )cos( x− y )

87

Precalculus Name:______________________________CW/HW- Trig Identities #3- Double and ½ angle

Date:______________________________


1. sin 4 x=2sin2 x cos2x 2.

4 tan x2−2 tan2 x

= tan2 x

3. (sin x+cos x )2=1+sin 2 x 4.csc 2 x=1

2sec x csc x

5. 2csc2 x=csc2 x tan x 6. cot x+tan x=2 csc2 x

7.cot x−tan x=4 cos2 x−2

sin 2 x 8. cot x− tan x=2cot 2 x

9.sec 2 x=sec2 x

2−sec2 x 10.cos2 x=1−tan2 x

1+tan2 x

88


89

Precalculus Name:______________________________Lesson: Sum, difference and ½ angle identities

Date:_______________________________

Objective: use the sum, difference, and half-angle identities to evaluate trigonometric expressions

RECALL-

Sum and Difference Identities

Half Angle Identities

Double Angle Identities

90

Showing all work, complete the following chart to find the exact value of each trigonometric expression using the specified trigonometric identity:

use a sum or difference identity use a half-angle identity

cos 105

sin 75

tan 165

91

PrecalculusName:__________________________________

Lesson- Solving Trig Equations I

Date:___________________________________

Objective: To learn how to solve basic trig equations using a variety of methods.-----------------------------------------------------------------------------------------------------------------------------------

DO NOW- sin 135o =

-----------------------------------------------------------------------------------------------------------------------------------Process:

EXACT Using GC (1) Using GC (2)

-----------------------------------------------------------------------------------------------------------------------------------Examples:

EXACT1. 2 cos x+1=00≤x<2 π

2. √3 tan+1=00≤x<π

92

3. √2sinθ−1=00≤θ<360 °

USING GC (1)

4. 5 cos x−2=00≤x<2 π

5. 4 tan θ+15=00≤θ<180 °

6. 5 .0118sin x−3 .1105=0 for all x

USING GC(2)

7. 2 x−cos x=0 for all x

93

8.tan2 x=1+3 x0≤x< π

4

EXIT- On separate paper

Solve: tan x+√3=00≤x<πPrecalculus Name:___________________________Lesson- Solving Trig Equations II

Date:_____________________________

Objective: To learn how to solve basic trig equations using a variety of methods.-----------------------------------------------------------------------------------------------------------------------------------

DO NOW- csc 135o =

-----------------------------------------------------------------------------------------------------------------------------------Process:

EXACT Using GC (1) Using GC (2)

------------------------------------------------------------------------------------------------------------------------------Examples:

EXACT

1.cos2 θ= 1

2sin 2 θ for all θ

94

2. cos x=cot x 0≤x<2 π

3. sin2 θ+2cos θ=−20≤θ<360 °

4. 4 cos2 2 x−4 cos2 x+1=00≤ x≤2 π

USING GC (1)

5. 4 cos2 θ=7 cosθ+20≤x<18095

6.8 sin2 θ+10 sinθ=30≤θ< π

2

USING GC(2)

7. 2 sin2 x=1−2sin x for all x

8. e−sin x=3−x for all x

PrecalculusName:____________________________

HW- Solving Trig Equations

Date:_____________________________

Solve each from 0≤θ≤2 πNOTE- You will need the graphing calculator to solve some but I won’t tell you which! That’s part of the fun!

1. 2 sin θ+5=4 sin θ+6 2. Tan2 θ=Tanθ

3.

2sin 2 θ2

=√32 4. cos2 θ=5+8cosθ

96

5. 2 sin2 θ−5sinθ+2=0 6. 3 sin2 θ−cosθ=1

7. 2 cosθ+1=secθ 8. sin2 θ+2=cosθ tan θ

9. sin2θ−1=0

97

Conics


Lesson- writing and using circle equations

Date:___________________________________

Objectives: write and use the standard and general form equations to graph circles

Do Now: Solve the following quadratic by completing the square:

2 x2−4 x+1=0

98

y

x

y

x

y

x

_______________________________________________________________________________________

General form of a conic: Ax2+Bxy+Cy2+Dx+ Ey+F=0

The different values of the coefficients determine which type of conic the equation forms.

Circles- locus of points equidistant from a single point. The general formula is as follows:( x−h )2+( y−k )2=r2

; where (h,k) is the center of the circle and r is the radius.

When the center of the circle is at the origin, the formula collapses down to:x2+ y2=r2

Example:Determine the values of the center and radius of the following circle equations.

(1) x2+ y2=8 (2) ( x−2 )2+ y2=25

_______________________________________________________________________________________

Writing Standard Form Equations & Graphing Circles:

(1) center at (3, -2) and radius of 3: (2) center at (2, 5) and radius of

√3 :

(3) center at (-3, 2) and tangent to the x-axis:

Writing General Form Equations for Circles:

99

Find the center and radius of the circle given each general form equation:

(1) x2 + y2 + 6x – 4y – 23 = 0

(2) x2 + y2 – 8x + 10y + 25 = 0

(2) x2+ y2−3 x+2 y+21=0


HW- Circle Equations

Date:___________________________________Using the given circle information:

100

(a) write the general form equation (b) graph the circle

(1) center at (0, 0) and radius of 4: (2) center at (-4, 2) and radius of 1: (3) center at (-1, -3) and radius of

√5 :

Using the given equation of a circle:

(a) find the center (b) find the radius

(4) (x – 5)2 + (y + 7)2 = 15 (5) x2 + y2 – 2x – 10y – 55 = 0

(6) x2 + y2 – 6x – 4y – 36 = 0 (7) x2 + y2 + 4x + 10y + 15 = 0


Lesson- write and use parabola equations

Date:___________________________________ Objectives: write and use the standard and general form equations to graph parabolas

101

Do Now: Write the equation of the circle given the center (2,3) and radius 12

Parabolas- locus of points equidistant from a fixed point and a fixed line, where the point is considered the focus and the line is considered the directrix. **Note the vertex is always ½ the distance between the focus and directrix

Let PF be the distance from any point on the parabola to the focus.PM be the distance from any point on the parabola to the directrix.

Rule: PF=PM

Therefore: ( x−h )2+( y−k )2=( y−d )2; where (h,k) is the focus and d is the horizontal directrixOr

( x−h )2+( y−k )2=( x−d )2; where (h,k) is the focus and d is the vertical directrix

To write the equation of a parabola give the focus and directrix, follow the steps below.1. Find the distance between the variable coordinate (x,y) and the focus using the distance formula.2. Find the distance between the variable coordinate (x,y) and the directrix using the distance formula..3. Set the result of step one equal to step two and solve for y in terms of x if directrix is horizontal or

solve for x in terms of y if directrix is vertical.

To determine the focus of an equation in standard form if horizontal directrix:Focus (h, k+a)

To determine the focus of an equation in standard form if vertical directrix:Focus (h+a, k)

Examples: Write the equation of the parabola given the following information.

1. Focus: (1,2) 2. Focus: (1,0)

Directrix: y=1

Directrix: x=2

Using the given information, find the equation of the parabola in standard form:

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1. focus at (2, 1), equation of the directrix is x = -2

2. vertex at (-1, 4), focus at (-1, 3)

3. focus at (0, 6), equation of axis of symmetry is y = 6, distance from focus to directrix is 3 to the left

4. focus at (4, -1), equation of the directrix is y = -5

_______________________________________________________________________________________5. vertex at (-5, 1), focus at (2, 1)

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6. focus at (7, -3), equation of the directrix is y = 1

7. vertex at the origin, focus at (0, 12)

8. vertex at the origin, equation of the directrix is x = -2

9. vertex at (-4, 5), focus at (-9, 5)

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Lesson- Graphing ellipse equations

Date:___________________________________

Objectives: write and use the standard form equation to graph ellipses

Do Now: Write the equation of the parabola given the focus: (2,2) and directrix: y=−2

Ellipse- the set of all points in a plane such that the sum of the distances from 2 fixed points (foci) is constant.

Standard form with center at (0,0):

x2

( Δx )2 + y2

( Δy )2=1;

where √( Δx )2

= horizontal change of vertices from center & √( Δy )2

= vertical change of vertices from center.

Standard form with center at (h,k) but also has its major axis on either axis:

( x−h )2

( Δx )2 +( y−k )2

( Δy)2 =1

Major axis: The line segment passing through the foci and intersecting the vertices.

Minor axis: The perpendicular bisector of the major axis

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To graph an ellipse:

Determine the center and vertices

Determine the location of the foci (Note: the distance from the end point of the minor axis to one of the foci is equal to ½ the distance of the major axis. Use the Pythagorean theorem.) ∴ √Big D−Little D=

distance from center to focus.

Draw a final sketch of the graph.

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Standard Form Equations & Graphing Ellipses:

Given the equation of an ellipse in standard form:(a) find the coordinates of the center, foci, and vertices(b) graph the ellipse (on a separate sheet of graph paper)

(1) x2

4+ y2

1=1

(2) ( x−1)2

100+( y+3 )2

5=1

(3) ( x−1)2

25+( y−3)2

16=1

(4) x2

6+ y2

18=1

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Precalculus Name:__________________________________HW- Graphing Ellipse Equations

Date:___________________________________

Using the given equation of an ellipse:108

(a) find the coordinates of the center, foci, and vertices (b) graph the ellipse

(1) x2

9+ y2

4=1

(2)

( x+4 )2

9+

( y+2 )2

25=1

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(3) x2

7+ y2

4=1

(4)

( x+3 )2

4+

( y−1 )2

16=1

Precalculus Name:__________________________________Lesson- Writing Ellipse Equations

Date:___________________________________

Objective: To learn how to write the equation of an ellipse in standard form

110

Writing the equation of an ellipse in standard form: Plot the given information Pretend that the major and minor axes are on the x and y axes unless told otherwise Find the vertices on the major axis and the intersection of the ellipse and the minor axis

Plug into

( x−h )2

( Δx )2 +( y−k )2

( Δy)2 =1; where Δx and Δy represent the distance from center to vertices and

(h,k) represent the center.

Using the given information, find the equation of the ellipse in standard form:

(1) center at (0, 0), length of vertical major axis is 20, length of minor axis is 12

(2) center at (2, 8), length of vertical minor axis is 16, distance of foci from center is 6

(3) foci on the line x = 4, minor axis on the line y = -3, length of major axis is 8, length of minor axis is 4

(4) center at (-3, -1), length of horizontal semi-major axis is 7, length of semi-minor axis is 5

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_______________________________________________________________________________________

(5) length of semi-major axis is 2√13 , foci at (-1, 1) and (-1, -5)

(6) center at (-5, -6), length of horizontal major axis is 10, distance between foci is 8

(7) length of minor axis is 6, foci at (3, 0) and (-3, 0)

(8) center at (3, -4), length of vertical major axis is 6, length of minor axis is 2

(9) length of major axis is 8, foci at (2, 0) and (-2, 0)

Precalculus Name:__________________________________Lesson: Graphing hyperbola equations

Date:___________________________________

Objective: write and use the standard form equation to graph hyperbolas

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Do Now: Sketch the graph of

x2

16+ y2

4=1

. Note the vertices, foci and lengths of major and minor axes.

Hyperbola: the set of all points in a plane such that the sum of the distances from 2 fixed points (foci) is constant.

( x−h )2

( Δx )2 −( y−k )2

( Δy )2 =1

or

( y−k )2

( Δy )2 −(x−h)2

( Δx)2 =1

Transverse axis: Distances from vertices of each part of the hyperbola

Conjugate axis: Perpendicular bisector of transverse axis

To graph an hyperbola:

Determine & plot the intercepts

Create a asymptote rectangle and use to create oblique asymptotes and determine equations

Use the Pythagorean theorem or √Big D + Little D=

”distance from center to focus” to find the location of the foci

This value added to and subtracted from the x-value of the center gives the foci

Draw a final sketch of the graph.

Standard Form Equations & Graphing Hyperbolas:

Given the equation of a hyperbola in standard form:(c) find the coordinates of the center, foci, and vertices

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(d) find the lengths of the transverse and conjugate axes(e) find the equations of the asymptotes(f) graph the hyperbola (on a separate sheet of graph paper)

(5) x2

16− y2

9=1

(6)

( y+3 )2

25−

( x−2 )2

16=1

(7) y2

4− x2

16=1

(8)

( x+2 )2

9−

( y−2 )2

25=1

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Precalculus Name:__________________________________HW- Graphing Hyperbola Equations

Date:___________________________________

Using the given equation of a hyperbola:

(c) find the coordinates of the center, foci, and vertices (d) find the equations of the asymptotes(e) find the lengths of the transverse and conjugate axes (f) graph the hyperbola

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(1) x2

25− y2

16=1

(2) y2

9− x2

4=1

(3)

( x−3 )2

4−

( y−4 )2

16=1

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Precalculus Name:_________________________________Lesson- Writing Hyperbola Equations

Date:__________________________________

Objective: To learn how to write the equation of a hyperbola in standard form.

Writing the equation of a hyperbola in standard form: Plot the given information

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Create an asymptote rectangle and diagonals. If vertical transverse axis, find the distance from the transverse and conjugate axes to the center and

plug into

( y−k )2

( Δy )2 −(x−h)2

( Δx)2 =1 for Δy and Δx , respectively. Plug in the center for (h,k).

If horizontal transverse axis, find the distance from the transverse and conjugate axes to the center and

plug into

( x−h )2

( Δx )2 −( y−k )2

( Δy )2 =1 for Δx and Δy respectively. Plug in the center for (h,k).

Using the given information, find the equation of the hyperbola in standard form:

(1) center at (0, 0), length of vertical transverse axis is 12, length of conjugate axis is 20

(2) center at (1, -4), length of horizontal transverse axis is 10, length of conjugate axis is 4

(3) foci at (1, -5) and (1, 1), length of transverse axis is 4

(4) vertices at (3, 4) and (3, 0), length of conjugate axis is 6

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(5) vertices at (0, 3) and (0, -3), a focus at (0, -9)

(6) center at (-3, 3), length of horizontal conjugate axis is 12, distance between foci is 12√2

(7) center at (-1, 8), length of vertical transverse axis is 6, distance between foci is 10

(8) foci at (6, 0) and (-4, 0), length of transverse axis is 8

(9) center at (-8, 4), length of vertical conjugate axis is 12, distance of foci from center is 9

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Precalculus Name:__________________________________Lesson- Identify Conic Sections

Date:___________________________________

Objective: To identify various conic sections including circles, ellipses, hyperbolas, and parabolas.

DO NOW: Transform the following quadratic to standard form:2 x2−4 x+1= y

_______________________________________________________________________________________

General form of a conic: Ax2+Bxy+Cy2+Dx+ Ey+F=0

Recall: The different values of the coefficients determine which type of conic the equation forms.

(1) y2 – 9x2 – 8y + 7 = 0

(2) y2 + 12x – 2y + 13 = 0

(3) x2 + 4y2 – 4x + 24y + 36 = 0

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(4) x2 – 4y2 + 6x – 8y – 11 = 0

(5) x2 – 2x – 8y + 17 = 0

(6) y−4=− 1

4( x+2 )2

(7) 9x2 + 16y2 – 32y – 128 = 0

(8) x2 + y2 – 6x + 10y – 47 = 0

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Precalculus Name:__________________________________CW/HW- Recognizing Conic Sections

Date:___________________________________

Objective: Practice determining the type of conic section based on the equation in general form

For each of the following:a. Write the equation in standard formb. Identify the conic sectionc. Find the necessary points and lines relevant to that conic sectiond. Graph the conic section

(1) 9x2 + 5y2 + 18x – 36 = 0

(2) x2 + y2 – 6x – 6y – 18 = 0

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(3) y2 – 4x + 6y + 25 = 0

(4) 4x2 – y2 – 8x + 6y – 9 = 0

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(5) x2 – y – 8x + 16 = 0

(6) 9x2 + 25y2 – 54x – 50y – 119 = 0

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(7) 16y2 – 9x2 + 36x – 32y – 164 = 0

(8) 5x2 + 2y2 – 40x – 20y + 110 = 0

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