Chapter 13

Rewrite each degree measure in radians and each radian measure in degrees. \(-15^{\circ}\)

Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=124^{\circ}, a=1, b=2 $$

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

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. \(A=56^{\circ}, B=22^{\circ}, a=12.2\)

Find the exact value of each trigonometric function. \(\sec 120^{\circ}\)

Rewrite each degree measure in radians and each radian measure in degrees. \(-225^{\circ}\)

Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=99^{\circ}, a=2.5, b=1.5 $$

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \cos \left(\operatorname{Arcsin} \frac{3}{5}\right) $$

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. \(a=4, b=8, c=5\)

Find the exact value of each trigonometric function. \(\tan 300^{\circ}\)

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{5 \pi}{6}\)

Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=33^{\circ}, a=2, b=3.5 $$

TRAVEL The cruise ship Reno sailed due west 24 miles before turning south. When the Reno became disabled and radioed for help, the rescue boat found that the fastest route to her covered a distance of 48 miles. The cosine of the angle at which the rescue boat should sail is \(0.5 .\) Find the angle \(\theta\) , to the nearest tenth of a degree, at which the rescue boat should travel to aid the Reno.

Find the exact value of each function. $$ \frac{\cos 60^{\circ}+\sin 30^{\circ}}{4} $$

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. \(a=21.5, b=13, C=38^{\circ}\)

Find the exact value of each trigonometric function. \(\cot 510^{\circ}\)

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{11 \pi}{4}\)

Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=68^{\circ}, a=3, b=5 $$

OPTICS You may have polarized sunglasses that eliminate glare by polarizing the light. When light is polarized, all of the wayes are traveling in parallel planes. Suppose horizontally-polarized light with intensity \(I_{0}\) strikes a polarizing filter with its axis at an angle of \(\theta\) with the horizontal. The intensity of the transmitted light \(I_{l}\) and \(\theta\) are related by the equation \(\cos \theta=\sqrt{\frac{1}{I_{0}}} \cdot\) If one fourth of the polarized light is transmitted through the lens, what angle does the transmission axis of the filter make with the horizontal?

Find the exact value of each function. $$ 3\left(\sin 60^{\circ}\right)\left(\cos 30^{\circ}\right) $$

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. \(A=40^{\circ}, b=7, a=6\)

Find the exact value of each trigonometric function. \(\csc 5400^{\circ}\)

Rewrite each degree measure in radians and each radian measure in degrees. \(-\frac{\pi}{4}\)

Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=30^{\circ}, a=14, b=28 $$

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\cot \left(\sin ^{-1} \frac{7}{9}\right)\)

Find the exact value of each function. $$ \sin 30^{\circ}-\sin 60^{\circ} $$

Find the exact value of each trigonometric function. \(\cos \frac{11 \pi}{3}\)

Rewrite each degree measure in radians and each radian measure in degrees. \(-\frac{\pi}{3}\)

Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=61^{\circ}, a=23, b=8 $$

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\cos \left(\tan ^{-1} \sqrt{3}\right)\)

Find the exact value of each function. $$ \frac{4 \cos 330^{\circ}+2 \sin 60^{\circ}}{3} $$

Find the exact value of each trigonometric function. \(\cot \left(-\frac{5 \pi}{6}\right)\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(225^{\circ}\)

A surveyor stands 100 feet from a building and sights the top of the building at a \(55^{\circ}\) angle of elevation. Find the height of the building.

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\tan (\operatorname{Arctan} 3)\)

Find the exact value of each function. $$ 12\left(\sin 150^{\circ}\right)\left(\cos 150^{\circ}\right) $$

Find the exact value of each trigonometric function. \(\sin \frac{3 \pi}{4}\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(30^{\circ}\)

In a sightseeing boat near the base of the Horseshoe Falls at Niagara Falls, a passenger estimates the angle of elevation to the top of the falls to be \(30^{\circ} .\) If the Horseshoe Falls are 173 feet high, what is the distance from the boat to the base of the falls?

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

A pilot typically flies a route from Bloomington to Rockford, covering a distance of 117 miles. In order to avoid a storm, the pilot first flies from Bloomington to Peoria, a distance of 42 miles, then turns the plane and flies 108 miles on to Rockford. Through what angle did the pilot turn the plane over Peoria?

Find the exact value of each function. $$ \left(\sin 30^{\circ}\right)^{2}+\left(\cos 30^{\circ}\right)^{2} $$

Find the exact value of each trigonometric function. \(\sec \frac{3 \pi}{2}\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(-15^{\circ}\)

RADIO A radio station providing local tourist information has transmitter on Beacon Road, 8 miles from where it intersects with the interstate highway. If the radio station has a range of 5 miles, between what two distances from the intersection can cars on the interstate tune in to hear this information?

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)\)

GEOMETRY A regular hexagon is inscribed in a unit circle centered at the origin. If one vertex of the hexagon is at \((1,0),\) find the exact coordinates of the remaining vertices.

Find the exact value of each trigonometric function. \(\csc \frac{17 \pi}{6}\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{3 \pi}{4}\)

FORESTRY Two forest rangers, 12 miles from each other on a straight service road, both sight an illegal bonfire away from the road. Using their radios to communicate with each other, they determine that the fire is between them. The first ranger's line of sight to the fire makes an angle of 38 with the road the second ranger's line of sight to the fire makes a \(63^{\circ}\) angle with the road. How far is the fire from each ranger?