MATHPOWER TM 12, WESTERN EDITION

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Chapter 5 Trigonometric Equations. 5.4. Trigonometric Identities. 5.4. 1. MATHPOWER TM 12, WESTERN EDITION. Trigonometric Identities. A trigonometric equation is an equation that involves at least one trigonometric function of a variable. The

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MATHPOWER TM 12, WESTERN EDITION

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Chapter 5 Trigonometric Equations

5.4

Trigonometric

Identities

5.4.1

MATHPOWERTM 12, WESTERN EDITION

Trigonometric Identities

A trigonometric equation is an equation that involves

at least one trigonometric function of a variable. The

equation is a trigonometric identity if it is true for all

values of the variable for which both sides of the

equation are defined.

Recall the basic

trig identities:

5.4.2

Trigonometric Identities

Quotient Identities

Reciprocal Identities

Pythagorean Identities

sin2q + cos2q = 1

tan2q + 1 = sec2q

cot2q + 1 = csc2q

sin2q = 1 - cos2q

tan2q = sec2q - 1

cot2q = csc2q - 1

cos2q = 1 - sin2q

5.4.3

Trigonometric Identities [cont’d]

sinx x sinx = sin2x

= cosA

5.4.4

Proving an Identity

Steps in Proving Identities

1. Start with the more complex side of the identity and work

with it exclusively to transform the expression into the

simpler side of the identity.

2. Look for algebraic simplifications:

• Do any multiplying , factoring, or squaring which is
• obvious in the expression.
• Reduce two terms to one, either add two terms or
• factor so that you may reduce.

3. Look for trigonometric simplifications:

• Look for familiar trig relationships.
• If the expression contains squared terms, think
• of the Pythagorean Identities.
• Transform each term to sine or cosine, if the
• expression cannot be simplified easily using other ratios.

4. Keep the simpler side of the identity in mind.

5.4.7

Proving an Identity

Prove the following:

a) sec x(1 + cos x) = 1 + sec x

= sec x + sec x cos x

= sec x + 1

1 + sec x

L.S. = R.S.

b) sec x = tan x csc x

c) tan x sin x + cos x = sec x

L.S. = R.S.

L.S. = R.S.

5.4.8

Proving an Identity

d) sin4x - cos4x = 1 - 2cos2x

1 - 2cos2x

= (sin2x - cos2x)(sin2x + cos2x)

= (1 - cos2x - cos2x)

= 1 - 2cos2x

L.S. = R.S.

e)

L.S. = R.S.

5.4.9

Proving an Identity

f)

L.S. = R.S.

5.4.10

Simplifying Trigonometric Expressions

Identities can be used to simplify trigonometric expressions.

Simplify.

b)

a)

5.4.5

Simplifing Trigonometric Expressions

c) (1 + tan x)2 - 2 sin x sec x

d)

5.4.6