Lesson 4.1 Polynomial Function

Objectives:

1. Identify a polynomial function from a given set of functions.2. Determine the degree of a polynomial function.

Study Guide:

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as in determinates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. It is a function that can be written in a form

P(x) = a0xn + a1xn-1 + a2xn-2 + … + an

P(x) has the following properties:

a0 – the first non-zero coefficient is called the leading coefficientn – the highest exponent is the degree

an – the constant term

Exercise 4.1

A. Identify which of the following are polynomial functions. Explain your answer.1. P(x) = 4x-22. P(x) = 2x

3. P(x) = 7x3+4x-3

4. f(x) =

5. g(x) = (x-3)3

6. h(x) = 10x+2

7. f(x) = 5+7x-3x2-4x3

8. P(x) = 25-4x2

9. f(x) =

10.h(x) =

B. For each function, determine the leading coefficient (LC), the leading term (LD), constant term (CT) and the degree (D) of each of the given polynomials.

LC LD CT Df(x) = 3x4+5x3+17x2-25x+10f(x) = 2x3+3x2-2x-5f(x) = x5-5x3+6x-2

4.2 Synthetic Division

Objectives:

1. Use synthetic division to find the quotient and the remainder wh a polynomial is divided by a linear expression of the form (x-c)

Study Guide:

Synthetic Division is also called Horner’s Method. Following are the steps in finding the quotient of P(x) divided by x-c using synthetic division:

a. Arrange the coefficients in descending powers of x in the first row, placing zeros for the missing terms.

b. Bring down the leading coefficient in the third row.

c. Multiply the entries in the third row by c, put the result in the second row under the next column, and add. Put the sum in the third row under the present column.

d. Repeat the third step until the column of the constant term is reached.

e. Write the quotient, Q(x), using the entries in the third row as the coefficient of the terms of x. The quotient is 1 degree lower than the degree of the dividend. The entry in the last column is the value of R(x).

Exercises 4.2

Divide P(x) by D(x) using long division then express P(x) in the form P(x) = Q(x)D(x)+R(x)

1. P(x) = x4+5x3-2x+8 D(x) = x-22. P(x) = 2x5-7x4+5x3-4x2-x+5 D(x) = x+13. P(x) = 3x5+5x4-3x3+x2+5x-2 D(x) = x+24. P(x) = 3x6+2x5-4x4+7x3-5x2+8x+1 D(x) = 3x-15. P(x) = 4x5-5x3+9 D(x) = 2x+3

Critical Thinking

What must be multiplied by x-1 to get x4-x3-3x2+10x-7? Explain your answer.

4.3 The Remainder Theorem and the Factor Theorem

Objectives:

1. State and illustrate the Factor Theorem

2. Find P(r) by synthetic Division and Remainder Theorem

Study Guide:

In Algebra, the polynomial remainder theorem is an application of polynomial long division. It states that the remainder of a polynomial f(x) divided by a linear divisor x-a is equal to f(a).

Exercises 4.3A. Answer the following questions below and put your answer to the corresponding by:*Note: To check your answer, make sure that the sum of vertical, horizontal and diagonal is equal to 15.

A. 7 B.

9 C. 1

D. 3 E.

A. (5x3+3x2-10x+2)/(x-2) 6 = 212B. (x4-5x3+7x2+9x-8)/(x-5) 2 = 34C. (x3-12x2+8x+60)/(x-10) 8 = -6D. (2x3+3x2-4x+5)/(x+2) 4 = 9E. (3x4-4x2+3x-2)/(x+1) 5 = -60

B. Determine if the given binomial is a factor of the given polynomial, then choose the letter of the correct answer.

1. f(x) = 5x4+16x3-15x2+8x+16 A. x-2

2. f(x) = x3+2x2-5x-6 B. x-3

3. f(x) = x5-2x4+3x3-6x2-4x+8 C. x+4

4. f(x) = 2x3+3x2-8x-12 D. x+1

5. x3-3x2+4x-12 E. 2x+3

C. Write YES f it is a factor and NO if it is not.

1. Is x-1 a factor of f(x) = 2x4+3x2-5x+7?

2. Is (2x-3)(x-1) are factors of f(x) = 2x2-5x+3?

3. Is (x+2) a factor of f(x) = x4-3x-5?

4. Is (x+3) a factor of 2x4+5x3-2x2+5x+3

Critical Thinking

1. Determine the value of k so that x-2 is a factor of x4+3x3+kx2+x-14?

2. If x-1 and x-2 are both factors of x4+Ax2-5x2+Bx+4, what are the values of A and B? Why?

4.4 Roots and their Multiplicities

Objectives:

1. Find the zeros of polynomial functions of degree greater than 2 using the factor theorem, synthetic division, Depressed Equations and Factoring.

Study Guide:

Example: Find the zeros of P(x) = (x-2)2(x+1)3(x+3)

Let (x-2)2(x+1)3(x+3)=0

Since the polynomial is already in factored form, then just equate each factor to zero and solve for x.

(x-2)2 = 0 (x+1)3 = 0 x+3 = 0

x-2 = 0 x+1 = 0 x = -3

x = 2 x = -1

It can be observed that x=2 is a zero of multiplicity of 2 of P(x) since (x-2)2 is a factor of P(x), while x=-1 is a zero of multiplicity 3 since (x+1)3 is also a factor of P(x). Therefore, the zeros of P(x) = (x-2)2(x+1)3(x+3) are 2 of multiplicity 2, -1 of multiplicity 3 and -3.Exercises 4.4A. Find all zeros of each polynomial function with their multiplicities.

1. P(x) = (x-3)(x+2)2

2. P(x) = (x+1)2(x-3)2(x+5)2

3. P(x) = x2(x+3)2(x-5)3

4. P(x) = (2x+3)2(3x-4)3(x+4)

5. P(x) = (x+4)2(x2-9)

6. P(x) = (2x-5)4(x2-2x-15)

7. P(x) = (4x-1)2(x-4)3(x2+5x-14)

8. P(x) = x3(5x+2)(2x2+5x+3)

9. P(x) = x(x-4)2(3x+2)3

10.P(x) = x2(3x-4)2(5x2-9x+4)

B. Use the factor theorem, synthetic division or factoring techniques to determine all the zeros of the following polynomial functions.

1. P(x) = x3+4x2-11x-30

2. P(x) = x4-x3-7x2+x+c

3. P(x) = x5-6x4+x3-40x2+16x

4. P(x) = x3-2x2-29x+42

5. P(x) = x4-5x3-14x2

Critical Thinking:

Determine the polynomial function of lowest integral coefficient if one of the zeros of

P(x) is . Justify your answer.

4.5 The Rational Roots Theorem and Descartes Rule of Signs

Objectives:

1. State and illustrate the Rational Zero Theorem2. Use Descartes’ Rule of Signs in finding the possible number of positive and

negative real zeros of a given polynomial function

Study Guide:The Rational Zero Theorem states that the possible rational zeros of a

polynomial must be equal to a factor p of the constant term divided by a factor of the leading coefficient. This means that the possible rational zeros are equal to p/q.Descartes’ Rule of Signs

1. The number of positive real zeros of f(x) is either equal to the number of sign changes or variations in signs of f(x) or is less than that number by an even integer. Note that if there is only one sign change in f(x), then f(x) has exactly one positive real zero.

2. The number of negative real zeros of f(x) is either equal to the number of sign changes or variations in signs of f(-x) or is less than that number by an even integer. Note that if f(-x) has only one sign change, then f(x) has exactly one negative real zero.

Exercises 4.5A. Determine all the possible zeros of the following polynomials.

1. P(x)=2x4-4x3-x2+4x-12. P(x)=4x3+11x2-2x+63. P(x)=4x4-3x3-10x2+3x-64. P(x)=2x5+2x4+3x3-3x2-6x-15. P(x)=5x6-45x4+14x3-90x+27

B. Make a chart summarizing the possible combinations of positive, negative and imaginary roots of each polynomial equations using Descartes’ Rule of Signs.

Number of Positive Real Roots

Number of Negative Real Roots

Number of Imaginary Roots

1. x2-x-5=02. x3-x2+3x+4=03. x3+4x2-x-12=04. 3x4+4x3-x2+x+4=0

5. 4x4-2x3+x2+2x-4=0

4.6 The Graphs of Polynomial Functions

Objective:

1. Draw the graphs of polynomial functions of degree greater than 2.

Study Guide:

The graph of polynomial function is a curve with turning points depending upon the degree of the polynomial.

Exercises 4.6

A. Prepare a table of values then sketch the graph of each polynomial function.

1. P(x)=x3+x2-5x+3

2. P(x)=-10-3x+6x2-x3

3. P(x)=x4-4x3-2x2+12x+9

4. P(x)=x4-8x3+22x2-24x+9

5. P(x)=8x2+4x3-2x4-x5

B. Communicating Mathematics

1. Based on the graphs in A, describe the graph of the polynomial function when:

a. The degree of the polynomial is even and an˃0.

___________________________________________________________________

b. The degree of the polynomial is even and an˂0.

___________________________________________________________________

c. The degree of the polynomial is odd and an˃0.

___________________________________________________________________

d. The degree of the polynomial is odd and an˂0.

___________________________________________________________________

2. At most, how many zeros does a polynomial functions have? Is it possible for a polynomial to have no zero? Explain.

CHAPTER 5: EXPONENTIAL AND LOGARITHMIC FUNCTIONS

5.1 The Nature of Exponential Functions

Objective:

1. Identify real-life relationships which are exponential in nature.

Study Guide:

A certain situation or occurrence shows exponential change of the original amount is multiplied by a fixed factor.

Exercises 5.1

A. Tell whether the situations show exponential change or not by completing the tables below.

1. A man buys a pair of signature pants worth P3000 with his credit card on condition that he pays his bill within a month or he will be charged an interest of 1% per month accumulated over the period he does not pay his bill.

Time (months) 0 1 2 3 4

Amount of Bill

2. A faucet leaks such that water drips from it at a fixed rate of 12 droplets per minute.

Time (minutes) 0 1 2 3 4 5

No. of Droplets 0 12

B. Analyze each situation and tell whether it is related to exponential change or not.

1. A new convenience store has initially 20 costumers and each week 2 new customers are coming.

2. A population of months increases by half of its population size every week.

3. If a student forgets to return a library book on the date it is due, he is fined P5 on the first day and P2 more each day thereafter.

4. The population of a certain type of microorganism doubles every hour.

5. A man accepts a position at P12000 a month with the understanding that he will receive P500 increase every year.

5.2 Other Exponential Functions

Objectives:

1. Describe some properties of the exponential function.2. Sketch the graph of an exponential function.

Study Guide:

Domain is the set of all first elements of the ordered pairs in a relation. Range is the set of all second elements of the ordered pairs in a relation. Intercept is the intersection of the graph with either the x-or-y-axis.

Exercises 5.2

A. 1. Consider a piece of string. Fold it into two then cut it. . Observe that there are now two pieces of string. Put the two pieces of string together, fold them again into two then cut the pieces. How many pieces of string are there now.

No. of folds (x) 1 2 3 4 5

No. of string y=f(x)

2.Draw the graph of the table above.

3.What mathematical sentence describes the relationship above?

4.What is the domain of the function?

5.Describe the graph obtained.

B. Sketch the graph of the exponential function of the form f(x)=ax , where a˃1 and x is a real number for the domain -3≤x≤3.

1. Draw the graphs of f(x)=2x, f(x)=3x and f(x)=4x on one set of axes.

x -3 -2 -1 0 1 2 3

f(x)=2x

f(x)=3x

f(x)=4x

2.What do the three graphs have in common?

C. Draw the graphs of f(x)=(1/2)x, f(x)=(1/3)x and f(x)=(1/4)x on one set of axes.

x -3 -2 -1 0 1 2 3

f(x)=(1/2)x

f(x)=(1/3)x

f(x)=(1/4)x

5.3 Solving Exponential Equations

Objectives:

1. Use the law of exponents to transfer exponential equations into algebraic equations.

2. Solve exponential equations.Study Guide:

Laws of Exponents

An exponential equation is an equation involving exponential functions. To solve an exponential equation, express both sides of the equation in the same base and solve for the value of the missing term.

A. Determine the solution/s to each exponential equation.

1. 3x=812x+5

2. 2x+1=32-2x+5

3. 162x-1=645x+3

4. 253x+1=125x+3

5. 7x+5=1/49

6. 22x=1/128

7. 3x^2+4x=1/27

8. 53x=125-x

9. 42x64x=1/512

10.125=(1/5)x+5

B. Solve for x.

1. 2.

3.

4.

5.

6.

C. Communicating Mathematics

1. Can you find a value of x such that f(x)=2x will be equal to zero? Justify your answer.

______________________________________________________________________

2. For any exponential function, f(x)=ax, is there any value of x so that ax=0? Explain your answer.

______________________________________________________________________

5.4 Inverse Functions

Objectives:

1. Define inverse functions2. Determine the inverse of a given function.

Study Guide:

Two functions f and g are inverse functions if and only if f(g(x))=x and g(f(x))=x. The inverse function of f is often denoted as f -1.

Exercises 5.4

A. Show the inverse of each of the following functions. Is the inverse still a function?

1. {(-2,4),(-1,1),(0,0),(1,1),(2,4)} __________

2. {(0,1),(1,2),(2,4),(3,8),(4,16)} __________

3. {(-2,-8),(-1,-1),(0,0),(1,1),2,8)} __________

B. Find the inverse of each relation.

1. f(x)= 5x-2

2. f(x)=

3. f(x)= 1/5(x-2)

4. f(x)=x2+2x-2-4

5. f(x)=

6. f(x)= 5(3x)

7. f(x)=

8. f(x)=

9. f(x)=

10. f(x)=

5.5 The Logarithmic Function

Objectives:

1. Define the logarithmic function as the inverse of the exponential function is the same base.

2. Describe the properties of logarithmic functions.

Study Guide:The mathematics of logarithms and exponentials occurs naturally in many

branches of science. It is very important in solving problems related to growth and decay. Therefore, we need to have some understanding of the way in which logs and exponentials work.

The formula y=logbbx is said to be written in logarithmic form and x=by is said to be written in exponential form. In working with these problems, it is most important to remember that y=logbx and x=by are equivalent statements.

Exercises 5.5Table Completion: Complete the table by converting the exponential to logarithmic form in Table A while convert the following logarithm to exponential form in Table B. Write the correct answer on the table.A.

Exponential Form Logarithmic Form

49½=7 1.

(1/9)-½=3 2.

23=8 3.

(1/5)-2=25 4.

42=16 5.

33=1/27 6.

(1/4)-1=4 7.

43/2=8 8.

zy=x 9.

64½=9 10.

B.

Logarithmic Form Exponential Form

log366=½ 1.

log82=1/3 2.

log 327=3 3.

log 1/749=-2 4.

log3/41=0 5.

log100.01=-2 6.

log2½=-1 7.

log10100=2 8.

log264=6 9.

log½1/8=3 10.

5.6 The Laws of Logarithms

Objective:

1. State and apply the laws of logarithms.

Study Guide:Four Basic Properties of Logarithms

1. logbxy = logbx + logby

2. logb = logbx-logby

3. logbxn = nlogbx4. logbx = logax/logab

Exercises 5.6A. Choose your best answer. Write your answer before the number. LETTER ONLY.

A. logarithm of product D. logarithm of quotient G. log7

B. Logarithm of root E. logarithm of power H. log72

C. logarithm F. log20 I. log6

1. It is logarithm that is equal to p times the logarithm of number.2. It is logarithm that is equal to the logarithm of the dividend minus the logarithm of

the divisor.3. log5+log4 = x4. log14-log2 = x5. It is logarithm that the two numbers equals the sum of the logarithms of the

numbers.6. log2+log3 = x7. What do call to another name for an exponent?

8. 2log3+3log2 = x9. It is logarithm that is equal to the logarithm of the number divided by r.

B. Direction: Express the following as single logarithms. Reach the star to get a bonus of 5 points. Write your answer on the box.1. log 5 + log 4=2. log 14 – log 2=3. 2log 3 + 3log 2=4. 2log 6 + log 2=

5.

6. log 7 + log 3=7. log 10 – log 2=8. 2log 10 – log 5=9. 4log 6 – 2log 2=10.3log 3 + 2log 5=

5.7 Common Logarithms

Objective:

1. Compute common logarithms using a:a. Calculatorb. Table of logarithms

Study Guide:

A logarithm to the base 10 is called the common logarithm. TO simplify the rotation needed to write them, we shall agree that when the base of a logarithm is not written, it is understood to be 10. That is:

log y = log10y

If log x = y, then x is called the antilogarithm of y. in symbols, x = antilog y.

Exercises 5.7

A. Between what two consecutive integers do each of the following logarithms lie? Do not use calculator.

1. log 236

2. log 178

3. log 67

4. log 3002

5. log 12000

6. log 0.03

7. log 2.3

8. log 0.005

9. log 1.1

10. log 7.5

B. Determine the value of each of the following common logarithms to six decimal places.

1. log 5

2. log 11

3. log 3.48

4. log 5.12

5. log 23.42

6. log 17.7

7. log 245

8. log 387

9. log 14320

10.

Communicating Mathematics

1. Your calculator displays an error when you try to find log 0. Why?

2. Why does your calculator display an error when you try to find log(-10)?

5.8 Solving Logarithmic Equations

Objectives:

1. Solve logarithmic equations.2. Use the laws of logarithms to solve logarithmic equations.

Study Guide:Logarithmic equations are equations involving logarithmic functions. To solve logarithmic equations, apply the laws of logarithms/exponents.

Exercises 5.8

A. Solve for the unknown

1. logx27 = 3

2. log2/3x = 2

3. loge20 = x

4. log1080 = x

5. log4(x+3) = 2

6. log (2x-1) = log (4x-3) – log x

7. log y = log 5x

8. log2x = 4.5

9. log23x = 4.5

10. log9x = 1

B. Loop the words that have a connection in logarithm.

Q U O T I E N T K E

D P O W E R Y Z R X

B T C U D O R P E P

K R L B Q M N C B O

L O G A R I T H M N

Z O E S P Z D J U E

L T A E U X F G N N

T R I A N G L E S T

5.9 Problem Solving with Logarithmic and Exponential Equations

Objective:

1. Solve problems involving logarithms and exponential equations.

Study Guide:

Exponential and logarithmic functions have many applications not only in science but also in business.

The use of calculator is a great help in solving many of the problems involving exponential and logarithmic functions.

Exercises 5.9

Solve the following problems.

1.

Principal Rate (%) Time in Years Compounded Amount

1. P60000 5 2 Every 6 mos.

2. P120000 10 1.5 Yearly

3. P240000 4 8 Every 2 mos.

4. P15000 2.5 5 Yearly

2. If P50000 is invested at 5% today, how much will it be worth at the end of 3 years if it is compounded

a. annually?

b. semi-annually?

c. quarterly?

d. monthly?

3. On her 7th birthday, Princess’ parents placed P20000 in time deposit at 5% interest compounded monthly. In ten years, how much money would be available for her educational expenses?

4. A certain city has a population of 2000 and a growth rate of 2.5%. What will be the expected population after 5 years?

5. If the half-life of a certain radioactive substance is 100 years, what fraction of the original amount of substance will remain after 400 years? after 600 years?

CHAPTER 6: Circular Functions

6.1 Measuring Angles in Radians

Objectives:

1. Convert angle measures from degrees to radians, and vice versa.2. Illustrate angles in standard position.3. Determine the coterminal angle or angles and the reference angle of an angle.

Study Guide:

A central angle whose arc is equal in length to the radius of the circle is called a radian. The radian measure of θ is defined to be the ratio of the arc length S to radius r:

where r is the radius, S is the arc length and θ is the measure of the angle in

radians.

If θ is a complete revolution, S=2πr and .

If θ is a complete revolution, the degree measure of θ is 360°.

Exercises 6.1

A. Radian. The Snowman.

The snowman picture has a unit circle for its base. You are to label 16 points in the unit circle with the radian measure inside the circle and the coordinates of the points outside the circle. Then color and decorate the snowman.

B. Wrute ‘True’ in the space provided if the underlined statement is correct, if not write ‘False’.

1. Radian is the ration between the length of an arc and its radius.

2. Radian is the standard unit of angular measure.

3. It is widely used in English.

4. The unit was formerly an SI complementary unit.

5. The radian is represented by the symbol “dian”.

6. Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc.

7. Radian is credited to Roger Cotes.

8. The term radian first appeared in print on June 5, 1873.

9. Radian is a false number.

10.The magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the length of the radius of the circle.

C. Convert the following Radians to Degrees.

1. 5π

2. 321π

3. 12π

4. 412π

5. 2 π

6.

7.

8.

9.

10.

D. Convert the following Degrees to Radians.

1. 0°

2. 30°

3. 45°

4. 60°

5. 90°

6. 120°

7. 135°

8. 150°

9. 180°

10.360°

6.2 The Sine and Cosine Functions

Objectives:

1. Define the sine and cosine functions of an angle θ, given a point P(θ) on the unit circle.

2. Evaluate the sine and cosine functions of special angles, and use identities to quickly derive the others.

Study Guide:Given a right triangle, ABC with its parts labeled, the ratio of the side opposite θ to the hypotenuse is called the sine of the measure of θ.

The ratio of the side adjacent θ to the hypotenuse is called the cosine of the measure of θ.

Exercise 6.2

A. Fill the table with the correct values of special angles.

θ sin θ cos θ tan θ sec θ csc θ cot θ

30°

45°

1

60°

2

90° 1 0 0

6.4 The Other Circular Functions

Objectives:

1. Define the four other circular functions.

2. Evaluate the circular functions of special angles

Study Guide:

We define the other four circular functions-tangent function, cosecant function, secant function and cotangent function in terms of the sine and cosine functions.

Exercise 6.4

A. Direction: Match the given trigonometric functions on column A with the given values on column B.

COLUMN A

1. cos 90°

2. csc 60°

3. sin 45°

4. cot 30°

5. sec 0°

6. sin 30°

7. sec 45°

8. sec 60°

9. tan 90°

10. csc 0°

COLUMN B

a. Undefined

b. ½

c. 1

d. 2

e.

f. 0

g.

h.

i.

B. Illustration

Direction: Draw, label and put description on the given angle of triangles.

1-3. 60°-30°-90°

4-6 45°-45°-90°

7-10. 30°-60°-90°

6.5 Graphs of Other Circular Functions

Objective:

1. Sketch the graphs of the circular functions.

Study Guide:

The secant and cosine functions are reciprocals. Hence the secant function can

be graphed by making use of the cosine since (cosθ≠0).

Exercise 6.5

1. Complete the table.

Function Domain Range

Sine

Cosine

Tangent

Cotangent

Secant

Cosecant

2. Complete the table.

Function Period Amplitude

y = 2 sin 3θ

y = ½ sin θ/2

y = 4 sin 5θ

y = 3 cos θ/3

y = ½ cos 4θ

3. Sketch the graph of each of the following functions on the interval 0 ≤ θ ≤ 2π. Give the period and amplitude of each function.

a. y = sin 3θ

b. y = cos θ/2

c. y = sec 2θ

d. y = csc 3θ

e. y = -cos θ

4. Sketch the graph of each of the following functions on the interval –π ≤ θ ≤ π.

a. y = tan 2π

b. y = ½ tan θ

c. y = 3 tan π

d. y = cot 2θ

e. y = -2 tan θ

6.6 The Fundamental Trigonometric Identities

Objective:

1. State the fundamental trigonometric identities and use them to prove other identities.

Study Guide:

A trigonometric identity is an equation involving trigonometric functions that can be solved by any angle. Trigonometric identities have less to do with evaluating functions at specific angles than they have to do with relationships between functions.

Reciprocal Identities:

Quotient Identities:

Pythagorean Identities:

sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 1 + cot2 θ = csc2 θ

Negative-Angle Identities:

sin(-θ) = -sin θ cos(-θ) = -cos θ tan(-θ) = -tan θ

Exercises 6.6:

A. For each trigonometric expression in Column A, choose the expression from Column B that completes a fundamental identity.

Column A

1.

2. tan x

3. cos(-x)

4. tan2 x + 1

5. 1

Column B

a. sin2 x + cos2 x

b. cot x

c. sec2 x

d.

e. cos x

B. Direction: Use the fundamental identities to get an equivalent expression then simplify it.

1. cot θ sin θ

2. cos θ csc θ

3. sin2 θ(csc2 θ – 1)

4. tan θ + cot θ

5. sin θ(csc θ – sin θ)

6. sin2 θ + tan2 θ + cos2 θ

7. sec θ cot θ sin θ

8. cot2 θ(1 + tan2 θ)

9. (sec θ – 1)(sec θ + 1)

10. (sec θ + csc θ)(cos θ – sin θ)

Lesson 6.7 Other Common Identities

Objective:

1. State other trigonometric identities.

Study Guide:

The following identities are derived from the fundamental trigonometric identities:

sin θ csc θ = 1

cos θ sec θ = 1

tan θ cot θ = 1

tan θ cos θ = sin θ

cot θ sin θ = cos θ

Exercises 6.7

Direction: Verify the following:

1.

2.

3. 1 + cot θ = csc θ(cos θ + sin θ)

4. sec4 θ – sec2 θ = tan4 θ + tan2 θ

Lesson 6.8 Double-Angle, Half-Angle, Sum-to-Product, and Product-to-Sum Formulas

Objectives:

1. Use the appropriate identities to evaluate expressions containing circular functions.

Study Guide:

Double-Angle Identities:

cos 2A = cos2 A – sin2 A

cos 2A = 1 – 2 sin2 A

cos 2A = 2 cos2 A – 1

sin 2A = 2 sin A cos A

Half-Angle Identities

Exercises 6.8

Directions: Use the identities to complete the following and simplify.

1.

2.

3. cos 14θ = 1 – 2 sin2 ____

4.

5.

6.

Lesson 6.9 Trigonometric Equations

Objective:

1. Solve simple trigonometric equations.

Study Guide:

Conditional equations with trigonometric (or circular) functions can usually be solved by using algebraic methods and trigonometric identities. For example:

2 sin θ + 1 = 0

2 sin θ = -1

sin θ = -½

Exercises 6.9

Directions: Solve each equation for solutions in the interval (0, 2π) by first solving for the trigonometric function.

1. 2 cot x + 1 = -1

2. 2 sec x + 1 = sec x + 3

3. 2 cos4 x = cos2 x

4. sin x + 2 = 3

5. tan2 x + 3 = 0

6. -2 sin2 x = 3 sin x + 1

7. 2 sin x + 3 = 4

8. sec2 x + 2 = -1

9. cos2 x + 2 cos x + 1 = 0

10. tan3 x = 3 tan x

Lesson 6.10 Functions Derived from the Sine and the Cosine Functions

Objective:

1. Describe the properties of functions derived from the sine and cosine functions.

Study Guide:

Sum and Difference Identities:

sin(A + B) = sin A cos B + cos A sin B

sin(A – B) = sin A cos B – cos A sin B

cos(A + B) = cos A cos B – sin A sin B

cos(A – B) = cos A cos B + sin A sin B

Exercises 6.10

Directions: Find the value of the following.

1. cos 15°

2. cos 75°

3. sin 105°

4. sin 165°

5. sin (-345°)