Chapter 10 - NCPN
Chapter 10 - NCPN
Chapter 10 - NCPN
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Lesson <strong>10</strong>.6 Verifying Trigonometric<br />
Identities<br />
Objectives<br />
Simplify a<br />
trigonometric<br />
expression.<br />
Verify a<br />
trigonometric<br />
identity.<br />
Trigonometric Identities<br />
The following trigonometric identities can be used to simplify<br />
expressions. These are equations that are true for all values of θ for<br />
which the expressions on each side of the equation are defined.<br />
Reciprocal Identities<br />
csc θ = 1 sec θ = 1 cot θ =<br />
1<br />
sin θ<br />
cos θ<br />
tan θ<br />
Tangent and Cotangent Identities<br />
sin θ<br />
cos θ<br />
tan θ = cot θ =<br />
cos θ<br />
sin θ<br />
Pythagorean Identities<br />
cos 2 θ + sin 2 θ = 1 1+ tan 2 θ = sec 2 θ<br />
1+ cot 2 θ = csc 2 θ<br />
Example 1 Simplifying a Trigonometric Expression<br />
Simplify the expression cos θ + sin θ tan θ.<br />
Solution<br />
Use the identity tan θ = sin θ<br />
cos θ .<br />
cosθ + sin θtan<br />
θ<br />
⎛ sin θ ⎞<br />
cosθ<br />
+ sin θ⎜<br />
cos θ ⎟<br />
⎝ ⎠<br />
2<br />
cos θ +<br />
sin θ<br />
cos θ<br />
⎛ 1 ⎞<br />
Multiply the expression by 1 in the form of ⎜ cos θ<br />
⎝ cos θ ⎟ .<br />
⎠<br />
⎛<br />
cos θ<br />
1 ⎞<br />
2<br />
cos θ<br />
sin θ<br />
⎜<br />
⎝ cos θ ⎠<br />
⎟ ⎛<br />
+ ⎞<br />
⎜ cos θ ⎟<br />
⎝<br />
⎠<br />
1 2 2<br />
(cos θ + sin θ)<br />
cos θ<br />
Because of the Pythagorean Identity cos 2 θ + sin 2 θ = 1, the result is<br />
1<br />
= sec θ.<br />
cos θ<br />
Ongoing Assessment<br />
Simplify the expression sin θ sec θ cot θ. 1<br />
<strong>10</strong>.6 Verifying Trigonometric Identities 471
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