Hwk 3 – Right Triangle Trig

Evaluate the following expressions. Your answer must be an angle in radians and in the interval $[-\frac{\pi }{2},\frac{\pi }{2}]$.

(a) ${\sin }^{-1}(-\frac{\sqrt{2}}{2})=$    

(b) ${\sin }^{-1}\left(\frac{1}{2}\right)=$    

(c) ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)=$    

Evaluate the following expressions. Your answer must be an exact angle in radians and in the interval $[0,\pi ]$. Example: Enter pi/6 for $\frac{\pi }{6}$.

(a) ${\cos }^{-1}\left(0\right)=$    

(b) ${\cos }^{-1}\left(\frac{\sqrt{2}}{2}\right)=$    

(c) ${\cos }^{-1}\left(\frac{1}{2}\right)=$    

Evaluate the following expressions. Your answer must be an exact angle in radians and in the interval $[-\frac{\pi }{2},\frac{\pi }{2}]$. Example: Enter pi/6 for $\frac{\pi }{6}$.

(a) ${\tan }^{-1}(-\sqrt{3})=$    

(b) ${\tan }^{-1}(-1)=$    

(c) ${\tan }^{-1}(-\frac{\sqrt{3}}{3})=$    

For the right triangle below, find the measure of the angle.
Figure is not to scale.

14<– ?11

    degrees

Evaluate the expression ${\sin }^{-1}\left(\cos \left(\frac{\pi }{4}\right)\right)$.

   

Give your answer as an exact value

Evaluate the expression ${\cos }^{-1}\left(\sin \left(\frac{11\pi }{6}\right)\right)$.

   

Give your answer as an exact value

Evaluate $\sin \left({\cos }^{-1}\left(\frac{1}{3}\right)\right)$, giving your answer as an exact value (no decimals)

Find a simplified expression for $\tan \left({\cos }^{-1}\left(\frac{b}{8}\right)\right)$

Find a simplified expression for $\sin \left({\tan }^{-1}\left(\frac{a}{8}\right)\right)$

Evaluate: ${\cot }^{-1}\left(\frac{-\sqrt{3}}{3}\right)$

Evaluate: ${\csc }^{-1}(-\sqrt{2})$

Evaluate: sin(arcsin$\left(\sqrt{7}\right)$)

Evaluate: arcsec$\left(\sec (-2\pi )\right)$

Evaluate: cot(arccos$\left(\frac{3}{14}\right)$)

Simplify the given expression.

$\cos \left({\sin }^{-1}\left(y\right)\right)=$    

Simplify the given expression.

$\sec \left({\tan }^{-1}\left(t\right)\right)=$    

For the right triangle below, find the measure of the angle in degrees. Round your answer to 2 decimal places.
Figure is not to scale.

10<– ?12

 degrees

Evaluate $\tan \left({\sin }^{-1}(-\frac{7}{10})\right)$, giving an exact answer.

A guy wire 1792 feet long is attached to the top of a tower. When pulled taut, it touches level ground 786 feet from the base of the tower. What angle does the wire make with the ground? Express your answer using degree measure rounded to one decimal place.

    degrees

Suppose $\sin \theta =\frac{7}{9}$, and $\theta$ is an angle in standard position.

Then the terminal side of $\theta$ could be in (check all that apply):

$\arcsin \left(\frac{7}{9}\right)$ is an angle whose terminal side is in (check all that apply):

Suppose $\cos \theta =-\frac{5}{8}$, and $\theta$ is an angle in standard position.

Then the terminal side of $\theta$ could be in (check all that apply):

$\arccos (-\frac{5}{8})$ is an angle whose terminal side is in (check all that apply):

Suppose $\tan \theta =-\frac{2}{9}$, and $\theta$ is an angle in standard position.

Then the terminal side of $\theta$ could be in (check all that apply):

$\arctan (-\frac{2}{9})$ is an angle whose terminal side is in (check all that apply):

Find $\arctan \left({\tan 220}^{\text{o}}\right)$.

$\arctan \left({\tan 220}^{\text{o}}\right)=$ degrees.

Find the exact value or state that it is undefined.  In the latter case, enter “DNE”.

${\cos }^{-1}\left(\cos (-\frac{3\pi }{4})\right)$    

 

Find the exact value or state that it is undefined.  In the latter case, enter “DNE”.

arcsec$\left(\sec \left(\frac{5\pi }{6}\right)\right)$    

 

Find the exact value or state that it is undefined.  In the latter case, enter “DNE”.

$\cot ($arccos$(-\frac{\sqrt{2}}{2}))$    

 

Find the exact value or state that it is undefined.  In the latter case, enter “DNE”.

$\cot ($arccos$\left(\frac{12}{13}\right))$    

 

Find the exact value or state that it is undefined.  In the latter case, enter “DNE”.

$\cot ($arccsc$\left(\frac{5}{4}\right))$    

 

A canyon is 7403 feet deep. From that point, a fire is seen at a location known to be 8 miles away from the base of the canyon wall. What is the angle of depression from the canyon edge to the fire? Round your answer to the nearest hundredths of a degree.

A para-sailor is being pulled by a boat on a lake. The cable is 194 feet long and the para-sailor is 190 feet above the surface of the water. what is the angle of elevation from the boat to the para-sailor? Round your answer to the nearest hundredths of a degree.