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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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slide1

Chapter 5 Trigonometric Equations

5.4

Trigonometric

Identities

5.4.1

MATHPOWERTM 12, WESTERN EDITION

slide2

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.

Prove that

Recall the basic

trig identities:

5.4.2

slide3

Trigonometric Identities

Quotient Identities

Reciprocal Identities

Pythagorean Identities

5.4.3

slide5

Simplifying Trigonometric Expressions

Identities can be used to simplify trigonometric expressions.

Simplify.

b)

a)

5.4.5

slide6

Simplifying Trigonometric Expressions

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

d)

5.4.6

slide7

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.

5.4.7

slide8

Proving an Identity

Prove the following:

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

1 + sec x

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

b) sec x = tan x csc x

5.4.8

slide9

Proving an Identity

d) sin4x - cos4x = 1 - 2cos2x

1 - 2cos2x

e)

5.4.9

slide11

Using Exact Values to Prove an Identity

Consider

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

b) Verify that this statement is true for x =

c) Use an algebraic approach to prove that the identity is true

in general. State any restrictions.

a)

5.4.11

slide12

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

b) Verify that this statement is true for x =

Rationalize the

denominator:

5.4.12

slide13

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

c) Use an algebraic approach to prove that the identity is true

in general. State any restrictions.

Restrictions:

Note the left side of the

equation has the restriction

Therefore,

where n is any integer.

The right side of the

equation has the restriction

Therefore,

And , where

n is any integer.

5.4.13

slide14

Proving an Equation is an Identity

Consider the equation

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

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

c) Use an algebraic approach to prove that the identity is true

in general. State any restrictions.

a)

5.4.14

slide15

Proving an Equation is an Identity [cont’d]

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

5.4.15

slide16

Proving an Equation is an Identity [cont’d]

c) Use an algebraic approach to prove that the identity is true

in general. State any restrictions.

Note the left side of the

equation has the restriction:

sin2A - sin A ≠ 0

sin A(sin A - 1) ≠ 0

The right side of the

equation has the restriction

sin A ≠ 0, or A ≠ 0.

Therefore, A ≠ 0, p + 2pn,

where n is any integer.

L.S. = R.S.

5.4.16

slide17

Assignment

Suggested Questions:

Pages 264 and 265

A 1-10, 21-25, 37,

11, 13, 16

B 12, 20, 26-34

5.4.16