Exam 8: Trigonometric Identities and Equations

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\cos \theta = \frac { 5 } { 6 }$ and $\theta$ is in quadrant IV.

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Determine all solutions of the equation in radians. -Find $\cos \frac { \theta } { 2 }$ , given that $\sin \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$ .

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. -Let $( a , b )$ and (c, d) be two points in the first quadrant, and let $\theta$ be the angle between the line segme connecting $( a , b )$ with the origin and the line segment connecting (c, d) with the origin. It can be shor $\cos \theta = \frac { a c + b d } { \sqrt { \left( a ^ { 2 } + b ^ { 2 } \right) \left( c ^ { 2 } + d ^ { 2 } \right) } }$ Find $\theta$ if $a = 4 , b = 2 , c = 8$ , and $d = 6$ . Give your answer in degrees rounded to the nearest hundredth.

Find the exact value of the expression using the provided information. -Find $\cos ( s + t )$ given that $\cos s = \frac { 1 } { 3 }$ , with $s$ in quadrant $I$ , and $\sin t = - \frac { 1 } { 2 }$ , with $t$ in quadrant IV.

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\sin 2 x + \sin x = 0$

Find the exact value by using a half-angle identity. - $\sin 22.5 ^ { \circ }$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Solve the equation (x in radians and $\boldsymbol { \theta }$ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - $\sin ^ { 2 } x + \sin x = 0$

Use identities to write the expression as a single function of x or $\theta$ - $\sin ( x + 2 \pi )$

Complete the sentence so the result is an identity. Let x be any real number. - $1 + \ldots = \csc ^ { 2 } x$

Solve the problem. -Suppose that a hair dryer operates on a voltage that can be represented by the relation $V ( t ) = 160 \cos ( 120 \pi t )$ and that it draws a current represented by the relation $I ( t ) = 12 \cos ( 120 \pi t )$ , where $t$ is time measured in seconds. The power consumed by the appliance is P =VI. Graph th 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 $\mathrm { P } = \mathrm { a } \cos ( \mathrm { k } \pi \mathrm { t } ) + \mathrm { d }$ , where $\mathrm { a } , \mathrm { k }$ , and $\mathrm { d }$ are constants.

Use a sum or difference identity to find the exact value. - $\sin 255 ^ { \circ }$

Verify that the equation is an identity. - $\sin ( x + y ) - \sin ( x - y ) = 2 \cos x \sin y$

Find the exact value of the real number y. - $y = \operatorname { arcsec } ( 1 )$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cot \theta$ if $\csc \theta = \frac { \sqrt { 10 } } { 3 }$ and $\theta$ is in quadrant $\mathrm { I }$ .

Solve the equation for x, where x is restricted to the given interval. - $3 y = \tan 6 x$ , for $x$ in $\left( - \frac { \pi } { 12 } , \frac { \pi } { 12 } \right)$

Write in terms of the cofunction of a complementary angle. - $\cos \frac { \pi } { 20 }$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\cos ( x + y ) = \cos x + \cos y$

Use a sum or difference identity to find the exact value. - $\tan \left( - \frac { 5 \pi } { 12 } \right)$