Exam 7: Trigonometric Identities and Equations

Use a sum or difference identity to find the exact value. - $\frac { \tan 155 ^ { \circ } - \tan 35 ^ { \circ } } { 1 + \tan 155 ^ { \circ } \tan 35 ^ { \circ } }$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$

Solve the problem. -The output voltage of a generator is given by $\mathrm { V } = 166 \cos \left( 2 \pi \mathrm { ft } - \frac { \pi } { 6 } \right)$ . Express the voltage as the sum of a sine and a cosine function.

Solve the problem. -In the study of alternating electric current, capacitive voltage is given by $\mathrm { V } _ { \mathrm { C } } = - \frac { \mathrm { i } } { 2 \pi \mathrm { f } } \cos 2 \pi \mathrm { ft } \text {, }$ where $\mathrm { f }$ is the number of cycles per second, $\mathrm { i }$ is the maximum current, and $\mathrm { t }$ is the time in seconds. Solve the equation for $\mathrm { t }$ .

Use the fundamental identities to find the value of the trigonometric function. -Find $\csc \theta$ if $\sin \theta = - \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\sin 2 \theta = \cos \theta$

Rewrite the following as a product of trigonometric functions. - $\sin \frac { \pi } { 13 } + \sin \frac { \pi } { 2 }$

Graph the inverse circular function. - $y=\cos ^{-1} x$

Find the exact value by using a half-angle identity. - $\sin 165 ^ { \circ }$

Solve the problem. -In the study of alternating electric current, capacitive voltage is given by $\mathrm { V } _ { \mathrm { C } } = - \frac { 1 } { 2 \pi \mathrm { f } } \cos 2 \pi \mathrm { ft }$ , where $\mathrm { f }$ is the number of cycles per second, $\mathrm { i }$ is the maximum current, and $\mathrm { t }$ is the time in seconds. Solve the equation for $\mathrm { t }$ .

Use identities to write the expression as a single function of x or θ. - $\tan \left( 30 ^ { \circ } + x \right)$

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

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

Write the word or phrase that best completes each statement or answers the question. -On the given graph of $y = \csc x$ sketch the graph of $y = \csc ^ { - 1 } x$ as defined in the text. Give the domain and the range.

Solve the equation for exact solutions. - $\arctan x = \arcsin \frac { 7 } { 25 }$

Solve the problem. -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 106 volts. Round values to 4 decimal places.

Use the fundamental identities to simplify the expression. - $\frac { \sin x \cos x } { \tan x }$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\cot \frac { \theta } { 3 } = 1$

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

Solve the equation for x, where x is restricted to the given interval. - $y = - \sec \frac { x } { 4 }$ , for $x$ in $[ 0,2 \pi ) \cup ( 2 \pi , 4 \pi ]$