Exam 7: Trigonometric Identities and Equations

Provide an appropriate response. -Explain what is wrong with the following solution for the equation $\sin 2 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$ . $\sin 2 \theta = \sqrt { 3 }$ $\sin \theta = \frac { \sqrt { 3 } } { 2 }$ $\theta = \frac { \pi } { 3 } \text { or } \theta = \frac { 2 \pi } { 3 }$

Find the exact value of the real number y. - $y = \csc ^ { - 1 } ( - 2 )$

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

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

Use an identity to write the expression as a single trigonometric function or as a single number. - $1 - 2 \sin ^ { 2 } 2 x$

Write in terms of the cofunction of a complementary angle. - $\cot \frac { 25 \pi } { 26 }$

Use a sum or difference identity to find the exact value. - $\sin 115 ^ { \circ } \cos 55 ^ { \circ } - \cos 115 ^ { \circ } \sin 55 ^ { \circ }$

Write the following as an algebraic expression in u, u > 0. - $\sin \left( \arcsin \frac { \mathrm { u } } { \sqrt { 3 } } \right)$

Solve the problem. -In an electric circuit, the electromotive force is defined as $V = \cos 2 \pi t$ , where $t$ is time in seconds. Solve the equation for $\mathrm { t }$ .

Solve the equation for solutions over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate. - $\tan 3 x - \sec 3 x = 5$

Solve the problem. -Two sound waves, producing the pressures $P _ { 1 } = 15 \sin ( 480 \pi t )$ and $P _ { 2 } = 8 \cos ( 480 \pi t )$ , combine on a person's ear drum to produce the pressure $P = P _ { 1 } + P _ { 2 }$ . Graph $P$ in $[ - 0.01,0.01,0.001 ]$ by $[ - 20,20,2 ]$ and use the graph to estimate values for $a$ and $k$ so that $P = a \sin ( 480 \pi t + k )$ . Round $k$ to the nearest ten-thousandth, if necessary.

Use the fundamental identities to simplify the expression. - $\cos x ( \csc x - \sec x ) - \cot x$

Use a sum or difference identity to find the exact value. - $\tan \frac { 7 \pi } { 12 }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cot \theta$ if $\tan \theta = \frac { \sqrt { 7 } } { 3 }$ and $\theta$ is in quadrant III.

Solve the equation for exact solutions. - $\arcsin \left( y - \frac { \pi } { 6 } \right) = \frac { \pi } { 6 }$

Solve the problem. -A rotating beacon is located a distance $\mathrm { d }$ from a long wall. The distance $\mathrm { d }$ is given by $\mathrm { d } = 8$ tan $2 \pi \mathrm { t }$ , where $t$ is the time measured in seconds since the beacon started rotating. Solve the equation for $t .$

Provide an appropriate response. -Graph the expression $\cos x + \sin x \tan x$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \cos ^ { - 1 } ( 0.8910 )$

Verify that each equation is an identity. - $( \sec \alpha - \tan \alpha ) ( \sec \alpha + \tan \alpha ) = 1$

Use the fundamental identities to simplify the expression. - $\cos x \tan x$