Chapter 13

Tom wants to build a rope bridge between his tree house and Roy’s tree house. Suppose Tom’s tree house is directly behind Roy’s tree house. At a distance of 20 meters to the left of Tom’s tree house, an angle of \(52^{\circ}\) is measured between the two tree houses. Find the length of the rope bridge.

Solve each equation by finding the value of \(x\) to the nearest degree. \(\operatorname{Arctan} 1=x\)

BASKETBALL The maximum height \(H\) in feet that a basketball reaches after being shot is given by the formula \(H=\frac{V_{0}^{2}(\sin \theta)^{2}}{64},\) where \(V_{0}\) represents the formula \(H=\frac{V_{0}^{2}(\sin \theta)^{2}}{64},\) where \(V_{0}\) represents the initial velocity and \(\theta\) represents the degree measure of the angle that the path of the basketball makes with the ground. Find the maximum height reached by a ball shot with an initial velocity of 30 feet per second at an angle of \(70^{\circ} .\)

Earth rotates on its axis once every 24 hours. How long does it take Earth to rotate through an angle of \(\frac{\pi}{6} ?\)

When landing, a jet will average a \(3^{\circ}\) angle of descent. What is the altitude \(x,\) to the nearest foot, of a jet on final descent as it passes over an airport beacon 6 miles from the start of the runway?

Solve each equation by finding the value of \(x\) to the nearest degree. $$ x=\operatorname{Arctan} \frac{\sqrt{3}}{3} $$

Find the exact value of each function. \(\sin 690^{\circ}\)

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(a=20, c=24, B=47^{\circ}\)

\(\cot \theta=-\frac{\sqrt{2}}{2},\) Quadrant \(\mathrm{IV}\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(60^{\circ}\)

Find the area of \(\triangle A B C\) to the nearest tenth. $$ A=60^{\circ}, b=12 \mathrm{cm}, c=12 \mathrm{cm} $$

Solve each equation by finding the value of \(x\) to the nearest degree. $$ x=\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right) $$

Find the exact value of each function. \(\cos 750^{\circ}\)

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(a=345, b=648, c=442\)

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((7,24)\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(425^{\circ}\)

Find the area of \(\triangle A B C\) to the nearest tenth. $$ C=136^{\circ}, a=3 \mathrm{m}, b=4 \mathrm{m} $$

Solve each equation by finding the value of \(x\) to the nearest degree. $$ x=\cos ^{-1} 0 $$

Find the exact value of each function. \(\cos 5 \pi\)

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(A=36^{\circ}, a=10, b=19\)

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((2,1)\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{\pi}{3}\)

Find the area of \(\triangle A B C\) to the nearest tenth. $$ B=32^{\circ}, a=11 \mathrm{mi}, c=5 \mathrm{mi} $$

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \cos ^{-1}\left(-\frac{1}{2}\right) $$

Find the exact value of each function. \(\sin \left(\frac{14 \pi}{6}\right)\)

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(A=25^{\circ}, B=78^{\circ}, a=13.7\)

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((5,-8)\)

Draw an angle with the given measure in standard position. \(235^{\circ}\)

Find the exact value of each function. \(\sin \left(-\frac{3 \pi}{2}\right)\)

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((4,-3)\)

Draw an angle with the given measure in standard position. \(270^{\circ}\)

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \operatorname{Arctan} \frac{\sqrt{3}}{3} $$

Find the exact value of each function. \(\cos \left(-225^{\circ}\right)\)

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(a=16, b=24, c=41\)

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((0,-6)\)

Draw an angle with the given measure in standard position. \(790^{\circ}\)

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \operatorname{Arccos} \frac{\sqrt{3}}{2} $$

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((-1,0)\)

Draw an angle with the given measure in standard position. \(380^{\circ}\)

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \sin \left(\sin ^{-1} \frac{1}{2}\right) $$

Two sides of a triangular plot of land have lengths of 425 feet and 550 feet. The measure of the angle between those sides is \(44.5^{\circ} .\) Find the perimeter and area of the plot.

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((\sqrt{2},-\sqrt{2})\)

Rewrite each degree measure in radians and each radian measure in degrees. \(120^{\circ}\)

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \cot \left(\sin ^{-1} \frac{5}{6}\right) $$

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(a=8, b=24, c=18\)

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((-\sqrt{3},-\sqrt{6})\)

Rewrite each degree measure in radians and each radian measure in degrees. \(60^{\circ}\)

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \tan \left(\cos ^{-1} \frac{6}{7}\right) $$

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(B=19^{\circ}, a=51, c=61\)

Find the exact value of each trigonometric function. \(\sin 240^{\circ}\)