Exam 8: Trigonometric Identities and Equations

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\cos 2 \theta = \frac { \sqrt { 3 } } { 2 }$

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

Use a calculator to give the value to the nearest degree. - $\theta = \tan ^ { - 1 } ( 2.2460 )$

Use a calculator to give the value to the nearest degree. - $\theta = \cos ^ { - 1 } ( 0.8910 )$

Express the function as a trigonometric function of x. - $\cos 4 \mathrm { x }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sqrt { \frac { 1 + \cos 22 ^ { \circ } } { 1 - \cos 22 ^ { \circ } } }$

Verify that each equation is an identity. - $\tan \theta \cdot \csc \theta = \sec \theta$

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

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. - $2 \sin ^ { 2 } x + \sin x = 1$

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\sqrt { 3 } \sec 2 \theta = 2$

Factor the trigonometric expression and simplify. - $1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x$

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

The statement tan-1(tan x) = x for all real numbers in the interval - $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$

Use Identities to find the exact value. - $\cos \left( - 75 ^ { \circ } \right)$

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 range $\mathrm { r }$ of a projectile is given by $r = \frac { 1 } { 32 } v ^ { 2 } \sin 2 \theta \text {, }$ where $\mathrm { v }$ is the initial velocity and $\theta$ is the angle of elevation. If $\mathrm { r }$ is to be $3000 \mathrm { ft }$ and $\mathrm { v } = 4000 \mathrm { ft } / \mathrm { sec }$ , what must the angle of elevation be? Give your answer in degrees to the nearest hundredth.

Graph the inverse circular function. - $y = \frac { 1 } { 2 } \operatorname { arccot } x$

Use the cofunction identities to find an angle $\theta$ that makes the statement true. - $\tan \left( 2 \theta - 140 ^ { \circ } \right) = \cot \left( \theta + 5 ^ { \circ } \right)$

Use the fundamental identities to simplify the expression. - $\cos ( - x ) \cos x - \sin ( - x ) \sin x$

Determine all solutions of the equation in radians. -Find $\cos \frac { \theta } { 2 }$ , given that $\cos \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $90 ^ { \circ } < \theta < 180 ^ { \circ }$ .

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