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of 24 Precalculus – Graphical, Numerical, Algebraic: Larson Chapter 4.1-4.6

Date: 4.1 Radian and Degree Measure

Syllabus Objective: 3.1 – The student will solve problems using the unit circle.

Trigonometry means the measure of triangles.

Standard Position (of an angle): initial side is on the positive x-axis;

positive angles rotate counter-clockwise

negative angles rotate clockwise

Radian: a measure of length; 1 radian when r s

r = radius, s = length of arc

s

r

Arc Length: s r ( must be measured in radians)

Recall: Circumference of a Circle; 2C r

So, there are 2 radians around the circle 2 radians = 360°, or radians = 180

Draw in the radians: Degrees and Radians:

Coterminal Angles: angles with the same initial and terminal sides, for example 360 or 2

Ex1: Find a positive and negative coterminal angle for each.

a. 130

b. 9

2

Initial side

Terminal side

r θ

s


Complementary Angles: The sum of two angles is 2

or 90

Supplementary Angles: The sum of two angles is or 180

Ex2: If possible find the complement and the supplement of

a.) 2

5

b.) 4

5

Converting Degrees (D) to Radians (R): 180

D R

(Note: 1180

)

Ex3: Convert 540° to radians.

Note: We multiply by 180

so that the degrees cancel. This will help you remember what to

multiply by.

Converting from Radians (R) to Degrees (D): 180

D R

Or use the proportion: 180

d r

Ex4: Convert 3

4

radians to degrees.

Note: We multiply by 180

so that the ’s cancel.


Reference Angle: every angle has a reference angle, with initial side as the x-axis

Ex.5

Draw and Find the reference angle of a.) 45 degrees, b.) 120 degrees, c.) 225 degrees, d.) 330 degrees.

Special Angles to Memorize (Teacher Note: Have students fill this in for practice.)

Degrees 30° 45° 60° 90° 120° 135° 150° 180°

Radians 6

4

3

2

2

3

3

4

5

6

Shortcut for Specials:

Degrees to radians

1. Find the reference angle

2. Divide reference angle into angle

3. Write reference angle in radians times answer in #2. 33011

330 30 times 116 6

Radians to degrees:

1. Write reference angle

2. Multiply example 4

4 times 60 2403

Reflection:

Date: 4.1 Radian and Degree Measure Continued Review:

Convert 1.2 hours into hours and minutes.

Solution:

Convert 3 hours and 20 minutes into hours.

Solution:

Babylonian Number System: based on the number 60; 360 approximates the number of days in a year.

Circles were divided into 360 degrees.

A degree can further be divided into 60 minutes (60'), and each minute can be divided into 60 seconds

(60'').


Converting from Degrees to DMS (Degrees – Minutes – Seconds): multiply by 60

Ex1: Convert 114.59° to DMS.

Converting to Degrees (decimal): divide by 60

Ex2: Convert 72° 13' 52'' to decimal degrees.

Note: The degree and minute symbols can be found in the ANGLE menu. The seconds symbol can be

found above the + sign (alpha +).

Arc Length: s r ( must be measured in radians)

Ex3: A circle has an 8 inch diameter. Find the length of an arc intercepted by a 240° central

angle.

Step One: Convert to radians.

Step Two: Find the radius.

Step Three: Solve for s.

Linear Speed (example: miles per hour): length

time

s rl

t t

Angular Speed (example: rotations per minute): angle

timeA

t

Ex4: Find the linear and angular speed (per second) of a 10.2 cm second hand.

Note: Each revolution generates 2 radians.

a.) Linear Speed: A second hand travels half the circumference in ____ seconds: or

b.) Angular Speed:

Note: The angular speed does not depend on the length of the second hand!

For example: The riders on a carousel all have the same angular speed yet the riders on the outside have a

greater linear speed than those on the inside due to a larger radius.


Ex5: Find the speed in mph of 36 in diameter wheels moving at 630 rpm (revolutions per

minute).

Unit Conversion:

1 statute (land) mile = 5280 feet Earth’s radius 3956 miles

1 nautical mile = 1 minute of arc length along the Earth’s equator

Ex6: How many statute miles are there in a nautical mile?

s r 3956r

Bearing: the course of an object given as the angle measured clockwise from due north

Ex7: Use the picture to find the bearing of the ship.

?

You Try

1. A lawn roller with 10 inch radius wheels makes 1.2 revolutions/second. Find the linear and

angular speed.

2. How many nautical miles are in a statute mile? Show your work.

Reflection:

QOD: How are radian and degree measures different? How are they similar?

N

51°

51°

N


Date: 4.2 Trigonometric Functions: The Unit Circle

Syllabus Objectives: Syllabus Objectives: 3.1 – The student will solve problems using the unit

circle. 3.2 – The student will solve problems using the inverse of trigonometric functions.

The coordinates x and y are two functions of the real variable t. You can use these coordinates to define

the six trigonometric functions of t.

sine cosecant

cosine secant

tangent cotangent

Let t be a real number and let (x,y) be the point on the unit circle corresponding to t.

1sin csc

1cos sec =

tan cot

t y ty

t x tx

y xt t

x y

The Unit Circle: 1r


Ex 1: Evaluate the six trigonometric functions at each real number.

a.) 6

t

b.) 5

4t

c.) 0t d.) t

You Try: Find the following without a calculator.

1. os

2. 5

cot3

3. sin225

Periodic Functions: A function y f t is periodic if there is a positive number c such that

f t c f t for all values of t in the domain of f.

The period of sine and cosine are 2 and the period of tangent is .

Ex2: Evaluate 3601

sin2

without a calculator.

Exploration: Consider an angle, θ, and its opposite, as shown in the coordinate grid. Compare the trig

functions of each angle.

sin , sin cos , cos tan , tan

csc , csc sec , sec cot , cot

**Cosine and secant are the only EVEN f x f x trig functions.

All the rest are ODD f x f x .

Sin t

Cos t

Tan t

Csc t

Sec t

Cot t

x

z

z

y

-y

θ −θ


Odd-Even Identities:

Ex3: Find a.) sin (-t) b.) cos (-t)

c.) Simplify the expression sin cscx x .

Evaluating Trigonometric Ratios on the Calculator

Note: Check the MODE on your calculator and be sure it is correct for the question asked (radian/degree).

Unless an angle measure is shown with the degree symbol, assume the angle is in radians.

Ex5: Evaluate the following using a calculator.

a. sin42.68 Mode: degree

sin42.68 0.678

b. sec1.2 Mode: radian

Note: There is not a key for secant. We must use 1/(cosx) because secant is the reciprocal of

cosine. sec1.2 2.760

Also recall: 1 1

csc and cotsin tan

x xx x

c. tan72 13 52 Mode: degree

Inverse Trigonometric Functions: use these to find the angle when given a trig ratio

Ex6: a.) Find θ (in degrees) if 5

cos12

.

1 5cos

12

Mode: degrees 65.376

b.) sin t = 1/3

REFLECTION: How is the unit circle used to evaluate the trigonometric functions? Explain.

sin sin and csc csc

tan tan and cot cot

cos cos and sec sec


Date: 4.3 Right Triangle Trigonometry

Syllabus Objectives: 3.1 – The student will solve problems using the unit circle. 3.2 – The student

will solve problems using the inverse of trigonometric functions. 3.5 – The student will solve

application problems involving triangles.

Note: The adjacent and opposite sides are always the legs of the right triangle, and depend upon which

angle is used.

Trigonometric Ratios of θ: Reciprocal Functions

Sine: opposite

sinhypotenuse

Cosecant: hypotenuse

cscopposite

Cosine: adjacent

coshypotenuse

Secant: hypotenuse

secadjacent

Tangent: opposite

tanadjacent

Cotangent: adjacent

cotopposite

Memory Aid: SOHCAHTOA

Think About It: Which trigonometric ratios in a triangle must always be less than 1? Why?

Ex1: For ABC , find the six trig ratios of A .

sin A cos A tan A

csc A sec A cot A

Note: It may help to label the sides as Opp, Adj, and Hyp first.

Find the six trig ratios for B .

sin A cos A tan A

csc A sec A cot A

Opposite

Adjacent Hypotenuse

10

8

6

B

A C

θ

Adjacent

Opposite Hypotenuse


Special Right Triangles: 30-60-90 & 45-45-90

Formulate Chart. This table must be memorized!

θ (Degrees) 30° 45° 60°

θ (Radians)

0 6

4

3

2

sinθ

cosθ

tanθ

4.3 Identities

Special Relationships: Identity: a statement that is true for all values for which both sides are defined

Example from algebra: 3 8 11 3 13x x

Exploration: Consider a right triangle.

Note that and are complementary. Write the trig functions for each angle. What do you notice?

sin cos cos sin tan cot

csc sec sec csc cot tan

**The trig functions of are equal to the cofunctions of θ, when and are complementary.

a

b

c

θ

Φ

x 3

30

°

x

2

x

45°

x

x 2

x

3x7


Cofunction Identities:

Ex2. a. sin 1 ˚=

. tan =

c. sec 25˚=

Ex3. cos α = .8 Find sin α and tan α using identities.

Reciprocal Identities

1 1 1sin cos tan

csc sec cot

1 1 1csc sec cot

sin cos tan

Quotient Identities sin cos

tan cotcos sin

Recall: Unit Circle 1, cos , sinr x y

Pythagorean Theorem: 2 2 1x y Pythagorean Identity: 2 2sin cos 1

To derive the other Pythagorean Identities, divide the entire equation by 2sin and then by 2cos :

sin 90 cos or sin cos2

cos 90 sin or cos sin2

sec 90 csc or sec csc2

csc 90 sec or csc sec2

tan 90 cot or cot tan2

cot 90 tan or cot tan2

,x y

Note: 22sin sin


Pythagorean Identities 2 2 2 2 2 2sin cos 1 1 cot csc tan 1 sec

Simplifying Trigonometric Expressions:

Look for identities

Change everything to sine and cosine and reduce

Ex4: Use basic identities to simplify the expressions.

a) 2cot 1 cos cos

cotsin

2 2 2 2

sin cos 1 sin 1 cos

b) tan csc sin

tancos

1csc

sin

Ex5: Simplify the expression

2

csc 1 csc 1

cos

x x

x

.

Use algebra: 2 2 2 21 cot csc cot csc 1

Application Problem

Ex6: A 6 ft 2 in man looks up at a 37 angle to the top of a building. He places his heel to toe 53

times and his shoe is 13 in. How tall is the building?

Draw a picture: The height of the building is (y + 74) inches.

Use trig in the right triangle to solve for y:

Height of the building in inches =

Convert to feet: The building is approximately _____________________ tall.

74''

53(13)=689''

y

37°

74''


You Try: Solve the triangle (find all missing sides and angles).

Reflection

QOD: Explain how to find the inverse cotangent of an angle on the calculator.

M

N

Q

q m

8

32°


Date: 4.4 Trigonometric Functions of Any Angle

Syllabus Objectives: 3.1 – The student will solve problems using the unit circle. 3.2 – The student

will solve problems using the inverse of trigonometric functions.

Trigonometric Functions of any Angle

Review: Memory Aid: To remember which trig functions are positive in which quadrant, remember

All Students Take Calculus. A – all are positive in QI, S – sine (and cosecant) is positive in QII, T –

tangent (and cotangent) is positive in QIII, C – cosine (and secant) is positive in QIV.

Review: Reference Angle: every angle has a reference angle, with initial side as the x-axis

Have Memorized:

Degrees 0°/360° 30° 45° 60° 90° 180° 270°

Radians 0 / 2

6

4

3

2

3

2

sinθ 0 1

2

2

2

3

2 1 0 −1

cosθ 1 3

2

2

2

1

2 0 −1 0

tanθ 0 3

3 1 3 und 0 und

1. Find the Quadrant (Use: All Students Take Calculus)

2. Find the Sign (positive or Negative)

3. Find Reference Angle

4. Evaluate (see chart above)

*See timed test of Trigonometric Functions.

2 2 2 2 2x y r r x y

sin csc

cos sec

tan cot

y r

r y

x r

r x

y x

x y

r

II (S)

sin (+)

tan (+)

III (T)

cos (+)

IV (C)

I (A)

sin (+)

cos (+)

tan (+) x

y

θ


Ex1: Find the six trig functions if 5

sin7

and tan 0 .

Since both sine and tangent are positive, we know that θ is in the ________ quadrant. Find the missing

leg (a): 2 2 25 7 _____________a a This is the adjacent leg of angle α.

5 oppsin

7 hyp

y

r

sin csc

cos sec

tan cot

Ex2: Find the sin of 8

3

Solving a Triangle: find the missing angles and sides with given information

Ex3: Solve the triangle (find all missing sides and angles).

4a (Pythagorean Triple: 3-4-5; or use the Pythagorean Thm)

5 7

α

A

B C

3

a

5


Evaluating Trig Functions Given a Point

Use the ordered pair as x and y

Find r: 2 2r x y

Check the signs of your answers by the quadrant

Ex4: Find the six trig functions of θ in standard position whose terminal side contains

point 5,3 .

Note: The point is in QII, so sine (and cosecant) will be positive.

sin csc

cos sec

tan cot

Ex5: Find the 6 trig functions of 330 .

To find the reference angle, start at the positive x-axis and go counter-clockwise 330 .

Reference Angle =____ ______, _____, ______x y r (30-60-90)

sin csc

cos sec

tan cot

Quadrantal Angles: angles with the terminal side on the axes, for example, 0 ,90 ,180 ,270

Needs to be memorized!

Degrees 0°/360° 30° 45° 60° 90° 180° 270°

Radians 0 / 2

6

4

3

2

3

2

sinθ

cosθ

tanθ

Ex6: Find csc13 .

Find the reference angle:

Reflection:

3

1


Date: 4.5 Graphs of Sine and Cosine Functions

Syllabus Objectives: 4.1 – The student will sketch the graphs of the six trigonometric functions. 4.3

– The student will graph transformations of the basic trigonometric functions. 4.6 – The student

will model and solve real-world application problems involving sinusoidal functions.

Teacher Note: Have students fill out the table as quickly as they can. Discuss patterns they can use to be

able to memorize these.

θ 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 360°

radians 0 6

4

3

2

2

3

3

4

5

6

7

6

5

4

4

3

3

2

2

sinθ

cosθ

tanθ

Sinusoid: a function whose graph is a sine or cosine function; can be written in the form

siny a bx c d

Sinusoidal Axis: the horizontal line that passes through the middle of a sinusoid

EX1 Graph of the Sine Function

Use the table above to sketch the graph.

a.) Radians: siny x b.) Degrees: siny

Sine is periodic, so we can extend the graph to the left and right.

Characteristics:

Domain: Range: y-intercept: x-intercepts:;

Absolute Max = Absolute Min = Decreasing:

Increasing: Period = Sinusoidal Axis:


EX 2 Graph of the Cosine Function

Use the table above to sketch the graph.

a.) Radians: cosy x b.) Degrees: cosy

Cosine is periodic, so we can extend the graph to the left and right.

Characteristics:

Domain: Range: y-intercept:

x-intercepts: Absolute Max = Absolute Min =

Decreasing: Increasing: Period = Sinusoidal Axis:

Note: We will work mostly with the graphs in radians, since it is the graph of the function in terms of x.

Recall: 2

f x a x h k is a transformation of the graph of 2f x x . a represents the vertical

stretch/shrink, h is a horizontal shift and k is a vertical shift.

Transformations of Sinusoids:

General Form siny a b x h k

a: vertical stretch and/or reflection over x-axis amplitude = a

h: horizontal shift phase shift = h k: vertical shift

period = 2

b

frequency =

2

b

b = number of cycles completed in 2

Amplitude: the distance from the sinusoidal axis to the maximum value (half the height of a wave)

Ex3: Sketch the graph of 4siny x .

Vertical stretch: Amplitude =


Period: the length of time taken for one full cycle of the wave 2

Pb

Frequency: the number of complete cycles the wave completes per unit of time freq = 2

b

Ex4: Sketch the graph of sin 2y x .

Period = Frequency =

Reflection: If 0a , the sinusoid is reflected over the _____axis.

Ex5: Sketch the graph of cosy x .

Note: Period, amplitude, etc. all stay the same.

Vertical Translation: in the sinusoid siny a b x h k , the line y k is the sinusoidal axis

Ex6: Sketch the graph of 4 siny x . Sinusoidal Axis:

Phase Shift: the horizontal translation, h, of a sinusoid siny a b x h k


Ex7: Sketch the graph of

cos2

y x

.

Shift cosy x to the right 2

. Does this graph look familiar? It is _______________________

Note: Every cosine function can be written as a sine function using a phase shift.

cos sin2

y x y x

Ex8: Sketch the graph of cos 4y x . Then rewrite the function as a sine function.

The coefficient of x must equal 1. Factor out any other coefficient to find the actual horizontal shift.

cos 4 cos 44

y x y x

Phase Shift: Period:

Graph:

Write as a sine function: cos 4y x

Writing the Equation of a Sinusoid

Ex9: Write the equation of a sinusoid with amplitude 4 and period 3

that passes through 6,0 .

Amplitude: ; Period: 2

3b

b

;

Phase Shift: normally passes through 0,0 , so shift right 6 units


You Try: Describe the transformations of 7sin 2 0.54

y x

. Then sketch the graph.

Reflection

QOD: How do you convert from a cosine function to a sine function? Explain.


Date: 4.6 Graphs of Other Trigonmetric Functions

Syllabus Objectives: 4.1 – The student will sketch the graphs of the six trigonometric functions. 4.3

– The student will graph transformations of the basic trigonometric functions.

θ 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 360°

radians 0 6

4

3

2

2

3

3

4

5

6

7

6

5

4

4

3

3

2

2

sinθ

cosθ

tanθ

Ex 1 Graph of the Tangent Function: Use the table above to sketch the graph.

a.) Radians: tany x b) Degrees: tany

Characteristics:

Domain: Range: Intercepts:

Increasing/Decreasing:

Period: Vertical Asymptotes:

Note: sin

tancos

xx

x , so the zeros of sine are the zeros of tangent, and the zeros of cosine are the vertical

asymptotes of tangent.

Ex 2 Graph of the Cotangent Function

Use the table above to sketch the graph. (Cotangent is the reciprocal of tangent.)

a.) Radians: coty x b.) Degrees: coty

Characteristics:

Domain: Range: Intercepts:

Increasing/Decreasing Period: Vertical Asymptotes:

Note: cos

cotsin

xx

x , so the zeros of cosine are the zeros of cotangent, and the zeros of sine are the vertical

asymptotes of cotangent.


Ex. 3 Graph of the Secant Function

Use the table above to sketch the graph. (Secant is the reciprocal of cosine.)

a.) Radians: secy x b.) Degrees: secy

Note: It may help to graph the cosine function first.

Characteristics:

Domain: Range: Intercept:

Local Max: Local Min: Period: Vertical Asymptotes:

Ex. 4 Graph of the Cosecant Function

Use the table above to sketch the graph. (Cosecant is the reciprocal of sine.)

a.) Radians: cscy x b.) Degrees: cscy

Note: It may help to graph the sine function first.

Characteristics:

Domain: Range: Intercept:

Local Max: Local Min: Period: Vertical Asymptotes:

Transformations: Sketch the graph of the reciprocal function first!

Ex5: Sketch the graph of 1 2secy x .

Graph 1 2cosy x . Shift up 1, amplitude 2. Use the zeros of the cosine function to determine the

vertical asymptotes of the secant function. Sketch the secant function.

Note: Amplitude is not applicable to secant.

The number 2 represents a vertical stretch.


Ex6: Sketch the graph of tan2

xy

.

Period: 21

2

P

Reflect over x-axis

Solving a Trigonometric Equation Algebraically

Ex7: Solve for x in the given interval using reference triangles in the proper quadrant.

sec 2,2

x x

Quadrant II; cos x Reference Angle = In QII:

Solving a Trigonometric Equation Graphically

Ex8: Use a calculator to solve for x in the given interval: cot 5, 02

x x

The equation cot 5x is equivalent to the equation 1

tan5

x . Graph each side of the equation and find

the point of intersection for 02

x

. Restrict the window to only include these values of x.

Note: We could have also graphed 1 2

1and 5

tany y

x . (Shown in the second graph above.)

Solution: _____________________________

You Try: Graph the function csc 2y x .

Reflection: QOD:

1. What trigonometric function is the slope of the terminal side of an angle in standard position?

Explain your answer using the unit circle.

2. Which trigonometric function(s) are odd? Which are even?

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