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MATHSTUDENT BOOK

12th Grade | Unit 5

Author: Alpha Omega Publications

Editors: Alan Christopherson, M.S. Lauren McHale, B.A.

Media Credits: Page 10: © Llepod, iStock, Thinkstock.

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2 |

ANALYTIC TRIGONOMETRY | Unit 5

Analytic Trigonometry

IntroductionIn this unit, algebra is used to study the relationships between the trigonometric functions. Trigonometric expressions are manipulated algebraically to simplify and evaluate them. Equivalent trig expressions and substitution are used in the process of solving trig equations.

Many of the applications connected with the material in this unit are seen in calculus or in sciences that require higher levels of math. Therefore, the unit focuses on logical thinking. Reasoning skills are developed through algebraic proof of trig identities.

ObjectivesRead these objectives. The objectives tell you what you will be able to do when you have successfully com-pleted this LIFEPAC®. When you have finished this LIFEPAC, you should be able to:

1. Simplify trigonometric expressions using fundamental trigonometric identities.

2. Determine equivalent trigonometric expressions using fundamental trigonometric identities.

3. Use trig identities to find the remaining trig function values of an angle when one value is known.

4. Solve trig equations using identities and substitution.

5. Determine equivalent trigonometric expressions using the sine, cosine, and tangent addition and subtraction formulas.

6. Evaluate trig functions using the sine, cosine, and tangent addition and subtraction formulas.

7. Evaluate trig functions of half-angle measures.

8. Determine equivalent trigonometric expressions using fundamental trigonometric identities.

9. Express a product of sine and cosine functions as a sum.

10. Express a sum of sine and cosine functions as a product.

11. Determine equivalent trigonometric expressions using product-to-sum and sum-to-product trigonometric identities.

Introduction | 3


Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here.

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4 | Introduction


1. IDENTITIES AND ADDITION FORMULAS

FUNDAMENTAL TRIGONOMETRIC IDENTITIESEquations that are true for all values of the variable are called identities. You may recall identities, such as x + 1 = 2x + 2, from your work in algebra. If an equation containing trig functions is true for all values in the domains of the functions, the equation is called a trigonometric identity.

Trigonometric identities are used to simplify expressions and calculate trig values. They also help you solve equations that would otherwise be impossible to solve.

Trigonometric identities are used extensively in calculus and in fields of study such as sound and optics.

Section ObjectivesReview these objectives. When you have completed this section, you should be able to:

• Simplify trigonometric expressions using fundamental trigonometric identities.

• Determine equivalent trigonometric expressions using fundamental trigonometric identities.

• Use trig identities to find the remaining trig function values of an angle when one value is known.

VocabularyStudy this word to enhance your learning success in this section.

identity . . . . . . . . . . . . . . . . . . . . . . . . . . . An equation that is true for all values of the variable in its domain.

Note: All vocabulary words in this LIFEPAC appear in boldface print the first time they are used. If you are not sure of the meaning when you are reading, study the definitions given.

RECIPROCAL IDENTITIESThe reciprocal identities are the definitions of the reciprocal functions:

csc θ = 1______sin θ , sin θ ≠ 0

sec θ = 1______cos θ, cos θ ≠ 0

cot θ = 1______tan θ, tan θ ≠ 0

In the following examples, the reciprocal identities are used to simplify expressions.

Keep in mind ...

Note that the functions are undefined when the denominator of the fraction is equal to zero. The identities hold true except for these values of θ.

Example

Simplify sin θ sec θ cos θ.

Solution

Use the reciprocal identity to substitute for sec θ:

sin θ sec θ cos θ = sin θ( 1______cos θ)cos θ

= sin θ(1)

= sin θ

Section 1 | 5


QUOTIENT IDENTITIESThe tangent and cotangent functions can be expressed in terms of sine and cosine.

Consider an angle, θ, in standard position. If a point on the terminal side of θ also lies on the unit circle (r = 1), then the coordinates (x, y) represent (cos θ, sin θ).

Using the fact that tan θ = opposite__________adjacent = y__

x and substitut-ing for x and y, you get

tan θ = sin θ______cos θ, cos θ ≠ 0

Since the cotangent function is the reciprocal of the tangent function, it can also be written in terms of sine and cosine: cot θ = 1______

tan θ

cot θ = cos θ______sin θ , sin θ ≠ 0

The quotient identities are

tan θ = sin θ______cos θ, cos θ ≠ 0

cot θ = cos θ______sin θ , sin θ ≠ 0

PYTHAGOREAN IDENTITIESThe Pythagorean identities are so named because they are derived from the Pythagorean theorem.

In the triangle shown, if T is on the unit circle, then x = cos θ, y = sin θ, and r = 1. The Pythagorean theorem gives us x2 + y2 = r2

cos2 θ + sin2 θ = 12


Two other identities can be derived from the previous identity.

T (x, y)

BO xθ

yr

Example

Simplify tan θ cos θ cot θ.

Solution

tan θ cos θ cot θ = tan θ cos θ( 1______tan θ)

= tan θ( 1______tan θ)cos θ

= (1)cos θ

= cos θ

Example

Simplify sin θ cot θ.

Solution

sin θ cot θ = sin θ(cos θ______sin θ )

= cos θ

Example

Simplify cos θ sin θ csc θ sec θ + sin θ tan θ cot θ.

Solution

cos θ sin θ csc θ sec θ + sin θ tan θ cot θ

= cos θ sin θ( 1______sin θ )( 1______

cos θ) + sin θ tan θ( 1______tan θ)

= cos θ( 1______cos θ)sin θ( 1______

sin θ ) + sin θ tan θ( 1______tan θ)

= 1(1) + sin θ = 1 + sin θ

T (x, y)

BO xθ

yr

6 | Section 1


Using the property of equality, divide the equation through by cos2 θ:

cos2 θ_______cos2 θ + sin2 θ_______

cos2 θ = 1_______cos2 θ

1 + tan2 θ = sec2 θ

Using the property of equality, divide the equation through by sin2 θ:

cos2 θ_______sin2 θ + sin2 θ_______

sin2 θ = 1_______sin2 θ

cot2 θ + 1 = csc2 θ

Thus, the Pythagorean identities are:

cos2 θ + sin2 θ = 11 + tan2 θ = sec2 θcot2 θ + 1 = csc2 θ

It is important to realize that algebraic properties of equality allow any identity to be written in other forms. In the following examples, the identities are “algebraically manipulated” to write equivalent expressions.

Identities are used to write trigonometric expressions in a simpler form. This might involve reducing the number of trig functions or eliminating a fraction.

Note that the original fraction is undefined when cos θ = 0. Therefore this identity holds true for all values of θ for which cos θ ≠ 0. So it is true when θ = π__

2 , 3π___2 , ...

Can you think of an alternate solution for the previous example?

Look at the following alternate solution and examine the differences and similarities in the two solutions.

In the next example, fractions are added together in order to simplify. Note that the Pythagorean identity is used to make a substitution in the numerator.

Examples

1. By subtracting cos2 θ from both sides of cos2 θ + sin2 θ = 1, you could write

sin2 θ = 1 − cos2 θ

2. Squaring both sides of the identity

tan θ = sin θ______cos θ results in

tan2 θ = sin2 θ_______cos2 θ

3. If cot θ = 1______tan θ, then

tan θ = 1______cot θ for cot θ ≠ 0

Example

Express sin θ sec θ_____________cos2 θ in terms of tan θ.

Solution

Look for the trig identities that have tangent in them: tan θ = sin θ______

cos θ

1 + tan2 θ = sec2 θ

Use substitution to replace the expressions in sin θ sec θ_____________

cos2 θ

Write the fraction as a product:

sin θ sec θ_____________cos2 θ = (sin θ______

cos θ)(sec θ______cos θ)

= (tan θ)(sec θ)( 1______cos θ)

Use sec θ = 1______cos θ to make a substitution:

= (tan θ)(sec θ)(sec θ) = tan θ sec2 θ

Use 1 + tan2 θ = sec2 θ to make a substitution: = tan θ(1 + tan2 θ) = tan θ + tan3 θ

É

Example

Express sin θ sec θ_____________cos2 θ in terms of tan θ.

SolutionWrite the fraction as a product:

sin θ sec θ_____________cos2 θ = (sin θ)(sec θ)( 1_______

cos2 θ)

Use sec θ = 1______cos θ and 1______

cos θ = sec2 θ to make substitutions:

= (sin θ)( 1_______cos2 θ)(sec2 θ)

= sin θ______cos θ(sec2 θ)

Use tan θ = sin θ______cos θ and 1 + tan2 θ = sec2 θ to

make substitutions: = tan θ(1 + tan2 θ) = tan θ + tan3 θ

Section 1 | 7


The identities can be used to determine all of the trig function values of an angle when one value is known. LET’S REVIEW

Before going on to the practice problems, make sure you understand all the main points of this lesson.

• If an equation containing trig functions is true for all values of the domains of the func-tions, the equation is called a trigonometric identity.

• Trig identities can be used to simplify trig expressions.

• There may be more than one approach to simplify a trig expression.

• Trig identities may be used to find the remaining trig values of an angle when one value is known.

Example

sin2 θ________cos θ + cos2 θ________

cos θ = sin2 θ + cos2 θ_________________cos θ

= 1______cos θ

= sec θ

Example

Find the five remaining trig function values of the second-quadrant angle, θ, if sec θ = - 3__

2 .

Solution

Cosine is the reciprocal of secant.

cos θ = 1______sec θ = - 2__

3


(- 2__3 )2 + sin2 θ = 1

4__9 + sin2 θ = 1

sin2 θ = 5__9

sin θ = ±√5___3

sin θ = √5___3 since the angle is in

Quadrant II.

csc θ = 1______sin θ = 3√5_____

5

tan θ = sin θ______cos θ = (√5___

3 ) ÷ (- 2__3 ) = -√5___

2

cot θ = 1______tan θ = - 2√5_____

5

Reminder:

Reciprocating √5___3 results in 3___

√5 which must be

rationalized:

3___√5

(√5___√5

) = 3√5_____5

8 | Section 1


Multiple-choice questions are presented throughout this unit. To enhance the learning process, students are encouraged to show their work for these problems on a separate sheet of paper. In the case of an incorrect answer, students can compare their work to the answer key to identify the source of error.

Complete the following activities.

1.1 Which of the following statements best describes a trigonometric identity? Select all that apply.

_______________________

a. An equation that holds true for all values of x. b. An equation that holds true for all values of y. c. An equation that holds true for all values of the domain. d. An equation that holds true for all values of the range.

1.2 Simplify the trigonometric expression sec (60) cos (60). _______

a. 1__2 b. 2 c. 1 d. √3___

2

1.3 Simplify sin θ______csc θ . _______

a. 1 b. cot θ c. csc2 θ d. sin2 θ

1.4 Simplify ( 1______csc θ )( 1______

sin θ ). _______

a. 1 b. csc2 θ c. sec2 θ d. sin2 θ

1.5 Simplify 1______cos θ + tan2 θ_______

cos θ . _______

a. 1 b. 1 + sin θ c. cos3 θ d. sec3 θ

1.6 Simplify cot θ tan3 θ + 1. _______

a. 2 b. csc2 θ c. sec2 θ d. tan2 θ

1.7 Find cot θ if θ terminates in Quadrant III and sec θ = -2. _______

a. ±√3 b. ±√3___3 c. √3___

3 d. √3

1.8 Simplify √_tan2 θ + 1___________cot2 θ + 1 . _______

a. 1 b. cot θ c. sec θ d. tan θ

1.9 Simplify csc2 θ + cot2 θ − 1. _______

a. 0 b. 2 c. cot2 θ d. 2 cot2 θ

Section 1 | 9


1.10 Simplify cot θ____________cos θ sec θ . _______

a. 1 b. cot θ c. cot2 θ d. tan θ

1.11 Simplify sin2 θ − sec θ cos θ + cos2 θ. ______________________

1.12 Simplify cos θ (tan θ + cot θ). _______

a. 1 b. cos2 θ c. csc θ d. sec θ

Match each trig function with its correct value if θ is an acute angle and csc θ = 2 1__2 .

1.13 _____________ √21____5

1.14 _____________ √21____2

1.15 _____________ 5√21______21

1.16 _____________ 2__5

1.17 _____________ 2√21______21

a. tan θ

b. cot θ

c. sin θ

d. cos θ

e. sec θ

10 | Section 1


804 N. 2nd Ave. E.Rock Rapids, IA 51246-1759

800-622-3070www.aop.com

MATHSTUDENT BOOK

ISBN 978-0-7403-3855-7

9 7 8 0 7 4 0 3 3 8 5 5 7

MAT1205 – Jul ‘18 Printing

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