Exam 6: Inverse Circular Functions and Trigonometric Equations

Solve the problem. -A rotating beacon is located a distance $d$ from a long wall. The distance $d$ is given by $\mathrm { d } = 4 \tan 2 \pi \mathrm { t }$ , where $t$ is the time measured in seconds since the beacon started rotating. Solve the equation for $t$ .

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

Provide an appropriate response. - $\text { True or false? The statement } \sin \left( \sin ^ { - 1 } x \right) = x \text { for all real numbers in the interval } \leq x \leq \pi \text {. }$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \cos ^ { - 1 } ( - 0.9397 )$

Write the following as an algebraic expression in u, u > 0. - $\sin ( \arctan u )$

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. - $\sin ^ { 2 } x + \sin x = 0$

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. - $\tan x ( \tan x - 2 ) = 6$

Solve the problem. -It can be shown that if the angle of elevation from an observer to the top of an object is A and the angle of elevation $\mathrm { d } \mathrm { ft }$ closer is $\mathrm { B }$ , then the distance from the object to the closest point of observation is given by $\mathrm { D } = \frac { \mathrm { d } \cot \mathrm { B } } { \cot \mathrm { A } - \cot \mathrm { B } } \mathrm { ft } \text {. }$ Find $B$ if $D = 20 \mathrm { ft } , \mathrm { d } = 30 \mathrm { ft }$ , and $A = 51 ^ { \circ }$ . Give your answer in degrees to the nearest hundredth.

Solve the equation for x, where x is restricted to the given interval. - $y = 7 \sin x \text {, for } x \text { in } \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Give the degree measure of . - $\theta = \arcsin \left( - \frac { 1 } { 2 } \right)$

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\tan x + \sec x = 1$

Give the exact value of the expression. - $\csc \left( \csc ^ { - 1 } \sqrt { 3 } \right)$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \tan ^ { - 1 } ( 0.5774 )$

Solve the problem. -The range r of a projectile is given by $\mathrm { r } = \frac { 1 } { 32 } \mathrm { v } ^ { 2 } \sin 2 \theta ,$ where $\mathrm { v }$ is the initial velocity and $\theta$ is the angle of elevation. If $r$ is to be $3000 \mathrm { ft }$ and $\mathrm { v } = 2000 \mathrm { ft } / \mathrm { sec }$ , what must the angle of elevation be? Give your answer in degrees to the nearest hundredth.

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $\sin 2 \theta = - \sin \theta$

Solve the problem. -The weekly sales in thousands of items of a product has a seasonal sales record approximated by $\mathrm { n } = 84.05 + 15.8 \sin \frac { \pi \mathrm { t } } { 24 }$ ( $\mathrm { t }$ is time in weeks with $\mathrm { t } = 1$ referring to the first week in the year). During which week(s) will the sales equal 91,950 items?

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. -Explain what is wrong with the following solution for the equation $\tan 3 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$ . $\tan 3 \theta = \sqrt { 3 }$ $\tan \theta = \frac { \sqrt { 3 } } { 3 }$ $\theta = \frac { \pi } { 6 } \text { or } \theta = \frac { 7 \pi } { 6 }$

Solve. -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.

Solve. -A formula for the up-and-down motion of a weight on a spring is given by $S = 2 \sin \left( \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \right) \mathrm { t }$ , where $\mathrm { k }$ is the spring constant, $\mathrm { m }$ is the mass, and $\mathrm { t }$ is the time. Solve the equation for $\mathrm { t }$ .

Provide an appropriate response. - $\text { True or false? The statement } \tan ^ { - 1 } ( \tan x ) = x \text { for all real numbers in the interval } - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } \text {. }$