Exam 8: Trigonometric Identities and Equations

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. -A generator produces an alternating current according to the equation $\mathrm { I } = 40 \sin 8 \pi \mathrm { t }$ , where $\mathrm { t }$ is time in seconds and $\mathrm { I }$ is the current in amperes. What is the smallest time $t$ such that $\mathrm { I } = 20$ ?

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. - $3 \cos ^ { 2 } \theta + 2 \cos \theta = 1$

Use identities to write the expression as a single function of x or $\theta$ - $\sin \left( \frac { \pi } { 4 } - x \right)$

Use the fundamental identities to simplify the expression. - $1 - \sin ^ { 2 } \theta$

Solve the equation for exact solutions. - $\arccos x + \arccos 2 x = \arccos \frac { 1 } { 2 }$

Use a sum or difference identity to find the exact value. - $\sin 25 ^ { \circ } \cos 95 ^ { \circ } + \cos 25 ^ { \circ } \sin 95 ^ { \circ }$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\sin ( \arctan 2 )$

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. -The output voltage for an AC generator is approximated by $\mathrm { v } = 156 \cos \left( 120 \pi \mathrm { t } - \frac { \pi } { 3 } \right)$ . Find the smallest positive value of $\mathrm { t }$ for which the output is 144 volts. Round values to 4 decimal places.

Use identities to find the indicated value for each angle measure. - $\cos \theta = \frac { 5 } { 13 } , \quad \sin \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Write the following as an algebraic expression in u, u > 0. -A painting 1 meter high and 3 meters from the floor will cut off an angle $\theta$ to an observer, where $\theta = \tan ^ { - 1 } \left( \frac { x } { x ^ { 2 } + 1.8 } \right)$ , assuming that the observer is $x$ feet from the wall where the painting is displayed and that the eyes of the observer are $1.8$ meters above the ground (see the figure). Find the value of $\theta$ for $x = 2$ . Round to the nearest tenth of a degree.

Perform the indicated operations and simplify the result so there are no quotients. - $\csc \theta ( \sin \theta + \cos \theta )$

Use an identity to write the expression as a single trigonometric function or as a single number. - $2 \cos ^ { 2 } 75 ^ { \circ } - 1$

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

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\tan ^ { 2 } \theta \csc ^ { 2 } \theta$

Write in terms of the cofunction of a complementary angle. - $\cot \frac { 17 \pi } { 18 }$

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. -A coil of wire rotating in a magnetic field induces a voltage given by e $= 20 \sin \left( \frac { \pi t } { 4 } - \frac { \pi } { 2 } \right)$ , where $t$ is time in seconds. Find the smallest positive time to produce a voltage of $10 \sqrt { 2 }$ . Round values to the nearest hundredth.

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\cos 2 x = \sin 2 x \tan x$

Solve the equation for exact solutions. - $- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }$

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

Find the exact value of the expression using the provided information. -Find $\sin ( s - t )$ given that $\cos s = \frac { 1 } { 3 }$ , with $s$ in quadrant $I$ , and $\sin t = \frac { 1 } { 4 }$ , with $t$ in quadrant II.