Exam 7: Trigonometric Identities and Equations

Solve the problem. -A generator produces an alternating current according to the equation $\mathrm { I } = 80 \sin 11 \pi \mathrm { t }$ , where $t$ is time in seconds and $I$ is the current in amperes. What is the smallest time $t$ such that $I = 40$ ?

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

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { 2 \tan 30 ^ { \circ } } { 1 - \tan ^ { 2 } 30 ^ { \circ } }$

Find the exact value of the real number y. - $y = \arcsin \left( \frac { 1 } { 2 } \right)$

Complete the sentence so the result is an identity. Let x be any real number. - $\frac { \sin x } { \tan x } =ـــــ$

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

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sin 8 x \cos 8 x$

Use identities to write the expression as a single function of x or θ. - $\sin \left( x + 45 ^ { \circ } \right)$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\csc x ( \sin x + \cos x )$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { 4 + 7 \tan x } { \sin x } = 4 \csc x + 7 \sec x$

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

Use a calculator to give the value to the nearest degree. - $\theta = \tan ^ { - 1 } ( 0.5774 )$

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

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

Use identities to find the indicated value for each angle measure. - $\cos 2 \theta = - \frac { 7 } { 9 }$ and $\frac { \pi } { 2 } < \theta < \pi \quad$ Find $\sin \theta$ .

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

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $\csc \theta = 1 + \cot \theta$

Give the exact value of the expression. - $\csc \left( \sin ^ { - 1 } \frac { 18 } { 30 } \right)$

Verify that each equation is an identity. - $\frac { 1 - \sec \theta } { \tan \theta } + \frac { \tan \theta } { 1 - \sec \theta } = - 2 \csc \theta$

Provide an appropriate response. -Explain what is wrong with the following solution for the equation $\cos 2 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$ . $\cos 2 \theta = \sqrt { 3 }$ $\cos \theta = \frac { \sqrt { 3 } } { 2 }$ $\theta = \frac { \pi } { 6 } \text { or } \theta = \frac { 11 \pi } { 6 }$