14.3 Verify Trigonometric Identities

Goal · Verify trigonometric identities.

Your Notes

VOCABULARY

Trigonometric identity

A trigonometric equation that is true for all values of q (in its domains)

FUNDAMENTAL TRIGONOMETRIC IDENTITIES

Reciprocal Identities

csc q = sec q = cotq =

Tangent and Cotangent Identities

tan q = cot q =

Pythagorean Identities

sin² q + cos² q = __1__

1+ tan² q =__sec² q__

1+cot² q =__csc²q__

Cofunction Identities	Negative Angle Identities
sin =__cos q__	sin (-q) =__-sinq__
cos =__sin q__	cos (-q) = __cosq__
tan =__cot q__	tan (-q) = __-tan q__

Your Notes

Example 1

Find trigonometric values

Given that cos q = and p < q < , find the values of the other five trigonometric functions of q.

Solution

Step 1 Find sin q.
sin ^2 q +cos ^2q =1	Write Pythagorean Identity.
sin2 q + =1	Substitute for cos q.
sin2q = 1-	Subtract from each side.
sin2 q = _____	Simplify.
sin q =	Take square roots of each side.

Because q is in Quadrant __III__, sin q is negative. So, sin q = -

Step 2 Find the values of the other four trigonometric functions of q using the known values of sin q and cos q.

tan q =

cot q =

csc q =

sec q =

Your Notes

Checkpoint Find the values of the other five trigonometric functions of q.

1. cos q = , 0 < q <

sin q = ; tan q = ; cot q = ;

csc q = ; sec q = 4

2. sin q = < q < 2p

cos q = ; tan q = ; cot q = ;

csc q = - 3; sec q =

Example 2

Simplify a trigonometric expression

Simplify the expression

Solution

= ______ · cot q	___Cofunction identity__
= _____ · ______	Cotangent identity
= _____	Simplify.
= __csc q__	__Reciprocal identity__

Checkpoint Simplify the expression.

3. · sin q + tan q csc q · cos³ q

1

Your Notes

Example 3

Verify a trigonometric identity

Verify the identity

sec q · - tan q cot q = .

Solution

sec q · - tan q cot q = sec q · __sec q__ - tan q cot q

= __sec² q_ - tan q cot q

= ___sec² q__ - tan q _{_____}

= __sec² q_ - __1__

= __tan² q__

= _{_______}

=

Checkpoint Verify the identity.

4.

=

= sin² x

Homework

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