MATHPOWERTM 12, WESTERN EDITION

5.4

5.4.1

Chapter 5 Trigonometric Equations

A trigonometric equation is an equation that involvesat least one trigonometric function of a variable. Theequation is a trigonometric identity if it is true for allvalues of the variable for which both sides of theequation are defined.

Trigonometric Identities

Prove that tan sincos

.

5.4.2

Recall the basictrig identities:

sin y

r

cos x

r

tan y

x

5.4.3

Trigonometric Identities

Quotient Identities

Reciprocal Identities

Pythagorean Identities

sinx x sinx =

cos 1

cos

cos 2cos

1

cos

cos 2 1

cos

sin A cos A 2 sin2 A 2sin Acos A cos 2 A

Trigonometric Identities [cont’d]

5.4.4

Identities can be used to simplify trigonometric expressions.

Simplifying Trigonometric Expressions

cos sin tana)

Simplify.

b)cot2

1 sin2

5.4.5

5.4.6

Simplifying Trigonometric Expressions

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

tan x cot x

5.4.7

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:

3. Look for trigonometric simplifications:

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

5.4.8

Proving an Identity

Prove the following:

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

1 + sec x

b) sec x = tan x csc x

sec x

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

sec x

d) sin4x - cos4x = 1 - 2cos2 x

1 - 2cos2x

e)

1

1 cos x

1

1 cosx 2 csc2 x

2 csc2 x

Proving an Identity

5.4.9

Proving an Identity

5.4.10

f)

cos A

1 sin A

1 sin A

cos A 2 sec A

Using Exact Values to Prove an Identity

5.4.11

Consider sin x

1 cos x

1 cosx

sinx.

b) Verify that this statement is true for x = 6

.

a) Use a graph to verify that the equation is an identity.

c) Use an algebraic approach to prove that the identity is true in general. State any restrictions.

a)

sin x

1 cos x

1 cosx

sinx

b) Verify that this statement is true for x = 6

.

Rationalize thedenominator:

Using Exact Values to Prove an Identity [cont’d]

5.4.12

c) Use an algebraic approach to prove that the identity is true in general. State any restrictions.

Using Exact Values to Prove an Identity [cont’d]

5.4.13

sin x

1 cos x

1 cosx

sinx

1 cos x

sin x

Note the left side of theequation has the restriction

Therefore, where n is any integer.

The right side of theequation has the restriction

Therefore, And , where n is any integer.

Restrictions:

Proving an Equation is an Identity

Consider the equationsin2 A 1

sin2 A sin A1

1

sin A.

b) Verify that this statement is true for x = 2.4 rad.

a) Use a graph to verify that the equation is an identity.

c) Use an algebraic approach to prove that the identity is true in general. State any restrictions.

a)

5.4.14

b) Verify that this statement is true for x = 2.4 rad.

Proving an Equation is an Identity [cont’d]

sin2 A 1

sin2 A sin A 1

1

sin A

5.4.15

5.4.16

Proving an Equation is an Identity [cont’d]

sin2 A 1

sin2 A sin A 1

1

sin A

c) Use an algebraic approach to prove that the identity is true in general. State any restrictions.

1 1

sin A

L.S. = R.S.

Note the left side of theequation has the restriction: sin2A - sin A ≠ 0

A 0, or A

2

Therefore, A 0 2 n or

A +2n, or

A 2

2 n, where n is

any integer.

The right side of theequation has the restrictionsin A ≠ 0, or A ≠ 0.

Therefore, A ≠ 0, + 2n,where n is any integer.

sin A(sin A - 1) ≠ 0

5.4.16

Suggested Questions:Pages 264 and 265A 1-10, 21-25, 37, 11, 13, 16B 12, 20, 26-34