Exam 6: Inverse Circular Functions and Trigonometric Equations

Solve the problem. -The optimal angle of elevation $\theta$ that a shot-putter should aim for in order to throw the greatest distance depends on the velocity $\mathrm { v }$ of the throw and the initial height $h$ of the shot. One model for $\theta$ that achieves this maximum distance is $\theta = \sin ^ { - 1 } \left( \sqrt { \frac { \mathrm { v } ^ { 2 } } { 2 \mathrm { v } ^ { 2 } + 64 \mathrm {~h} } } \right)$ . Suppose a shot-putter can consistently release the steel ball with velocity of 40 feet per second from an initial height $h$ of $5.7$ feet. What angle, to the nearest degree, will maximize the distance?

Write the following as an algebraic expression in u, u > 0. - $\tan \left( \cos ^ { - 1 } \frac { \mathrm { u } } { 3 } \right)$

Find the exact value of the real number y. - $y = \arccos \left( \frac { \sqrt { 3 } } { 2 } \right)$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\cos \left( \cos ^ { - 1 } ( - 0.9372 ) \right)$

Solve the problem. -Snell's Law states that $\frac { \sin \theta _ { 1 } } { \sin \theta _ { 2 } } = \frac { \mathrm { v } _ { 1 } } { \mathrm { v } _ { 2 } }$ where $v _ { 1 } , v _ { 2 }$ are the speeds at which light travels through two different mediums, and $\theta _ { 1 } , \theta _ { 2 }$ are the angles of incidence and refraction respectively. Find the angle of refraction if $\frac { \mathrm { v } _ { 1 } } { \mathrm { v } _ { 2 } } = 1.6$ and the angle of incidence is $26 ^ { \circ }$ . Give your answer in degrees to the nearest hundredth.

Solve the equation for solutions over the interval [ $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. - $\sin \frac { x } { 2 } + \cos \frac { x } { 2 } = 0$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \sin ^ { - 1 } ( - 0.4848 )$

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to one decimal place. -The equation $\sin 2 x - \cos 3 x = - 4$ has no solution in the interval $[ 0,2 \pi )$ . Explain what this tells you about the graph of $y = \sin 2 x - \cos 3 x + 4$ .

Solve the equation for solutions over the interval [ $[ 0,2 \pi ) .$ ). Write solutions as exact values or to four decimal places, as appropriate. - $\sin \frac { x } { 2 } + \cos \frac { x } { 2 } = \sqrt { 2 }$

Solve the problem. -A generator produces an alternating current according to the equation $\mathrm { I } = 16 \sin 88 \pi \mathrm { t }$ , where $t$ is time in seconds and $\mathrm { I }$ is the current in amperes. What is the smallest time $\mathrm { t }$ such that $\mathrm { I } = 8$ ?

Give the exact value of the expression. - $\cos \left( \arcsin \frac { 1 } { 4 } \right)$

Find the exact value of the real number y. -y = arcsec ( )

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\cos ^ { 2 } \frac { \theta } { 2 } = 1$

Solve the equation (x in radians and 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. - $4 \sin ^ { 2 } x - 1 = 0$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \tan ^ { - 1 } ( - 0.7002 )$

Graph the inverse circular function. - $y = \operatorname { arcsec } 3 x$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi ) \text {. }$ - $\sin 4 x = \frac { \sqrt { 3 } } { 2 }$

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

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

Give the degree measure of . -ϴ = arccos (0)