Exam 5: Trigonometric Identities

Use the fundamental identities to find the value of the trigonometric function. -Suppose that a household appliance draws a current represented by the relation $\mathrm { I } ( \mathrm { t } ) = 7 \cos ( 120 \pi \mathrm { t } )$ , where $t$ is time measured in seconds. The power consumed by the appliance is $P = I ^ { 2 } R$ , where $R$ is a constant. Take $R$ to be 16 and graph the power in [0,0.04, 0.01] by [-200, 2000, 200] and use an identity to write the expression for the power in the form $P = a \cos ( k \pi t ) + d$ , where $a$ , $k$ , and $d$ are constants.

Find the exact value of the expression using the provided information. - $\frac { 2 \tan 15 ^ { \circ } } { 1 - \tan ^ { 2 } 15 ^ { \circ } }$

Rewrite the following as a product of trigonometric functions. -Find $\sin ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = \frac { 1 } { 3 }$ , with $\mathrm { s }$ in quadrant $\mathrm { I }$ , and $\sin \mathrm { t } = \frac { 1 } { 4 }$ , with $\mathrm { t }$ in quadrant II.

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sin 22.5 ^ { \circ }$

Find the exact value by using a half-angle identity. - $\frac{\sin x}{\tan x}=$ -----

Verify that each equation is an identity. -tan x(csc x - sin x)= cos x

Complete the sentence so the result is an identity. Let x be any real number. -cos 255°

Find the exact value by using a half-angle identity. - $\tan 345 ^ { \circ }$

Use Identities to find the exact value. - $\sin \left( - \frac { 5 \pi } { 12 } \right)$

Use identities to find the indicated value for each angle measure. - $\cos \theta = - \frac { 5 } { 13 } , \frac { \pi } { 2 } < \theta < \pi \quad$ Find $\cos ( 2 \theta )$

Use the fundamental identities to simplify the expression. - $\frac { \sec ^ { 2 } x } { \tan x } - \tan x$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \frac { x } { 2 }$ , given that $\sin x = \frac { 1 } { 4 }$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$ .

Write in terms of the cofunction of a complementary angle. - $\cos \left( \frac { \pi } { 12 } \right)$

Verify that each equation is an identity. - $\sin ( 4 x ) = ( 4 \sin x \cos x ) \left( 2 \cos ^ { 2 } x - 1 \right)$

Find the exact value by using a half-angle identity. -cos 22.5°

Tell whether the statement is true or false. - $\cos - 23 ^ { \circ } = \cos 40 ^ { \circ } \cos 63 ^ { \circ } + \sin 40 ^ { \circ } \sin 63 ^ { \circ }$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. -cos x tan x

Find the exact value by using a half-angle identity. -(csc ϴ + cot ϴ)(1 - cos ϴ)= sin ϴ

Find the exact value by using a half-angle identity. - $\tan \frac { \pi } { 3 } = ----- \frac { \pi } { 6 }$

Verify that the equation is an identity. - $\frac { \cos ( x - y ) } { \cos ( x + y ) } = \frac { 1 + \tan x \tan y } { 1 - \tan x \tan y }$