Chapter 6

\(31-36\) me measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ 70^{\circ}, 430^{\circ} $$

\(31-36\) me measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ -30^{\circ}, \quad 330^{\circ} $$

9–32 Find the exact value of the trigonometric function. $$\sin \frac{11 \pi}{6}$$

33–36 Find the quadrant in which \(\theta\) lies from the information given. \(\sin \theta<0 \quad\) and \(\quad \cos \theta<0\)

\(31-36\) me measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ \frac{5 \pi}{6}, \frac{17 \pi}{6} $$

Distance Across a Lake Points \(A\) and \(B\) are separated by a lake. To find the distance between them, a surveyor locates a point \(C\) on land such that \(\angle C A B=48.6^{\circ} .\) He also measures \(C A\) as 312 \(\mathrm{ft}\) and \(C B\) as 527 ft. Find the distance between \(A\) and \(B\)

33–36 Find the quadrant in which \(\theta\) lies from the information given. \(\tan \theta<0 \quad\) and \(\quad \sin \theta<0\)

\(31-36\) me measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ \frac{32 \pi}{3}, \frac{11 \pi}{3} $$

Three circles of radii 4, 5, and 6 cm are mutually tangent. Find the shaded area enclosed between the circles.

The Leaning Tower of Pisa The bell tower of the cathedral in Pisa, Italy, leans \(5.6^{\circ}\) from the vertical. A tourist stands 105 \(\mathrm{m}\) from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of the tower to be \(29.2^{\circ} .\) Find the length of the tower to the nearest meter.

33–36 Find the quadrant in which \(\theta\) lies from the information given. \(\sec \theta>0 \quad\) and \(\quad \tan \theta<0\)

\(31-36\) me measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ 155^{\circ}, \quad 875^{\circ} $$

Prove that in triangle \(A B C\) \(a=b \cos C+c \cos B\) \(b=c \cos A+a \cos C\) \(c=a \cos B+b \cos A\) These are called the Projection Laws. [Hint: To get the first equation, add the second and third equations in the Law of cosines and solve for \(a\).]

Radio Antenna A short-wave radio antenna is supported by two guy wires, 165 \(\mathrm{ft}\) and 180 \(\mathrm{ft}\) long. Each wire is attached to the top of the antenna and anchored to the ground, at two anchor points on opposite sides of the antenna. The shorter wire makes an angle of \(67^{\circ}\) with the ground. How far apart are the anchor points?

\(31-36\) me measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ 50^{\circ}, \quad 340^{\circ} $$

33–36 Find the quadrant in which \(\theta\) lies from the information given. \(\csc \theta>0 \quad\) and \(\quad \cos \theta<0\)

Height of a Tree \(A\) tree on a hillside casts a shadow 215 ft down the hill. If the angle of inclination of the hillside is \(22^{\circ}\) to the horizontal and the angle of elevation of the sun is \(52^{\circ},\) find the height of the tree.

37–42 Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. \(\tan \theta, \quad \cos \theta ; \quad \theta\) in quadrant III

\(37-42\) me Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 733^{\circ} $$

A parallelogram has sides of lengths 3 and \(5,\) and one angle is \(50^{\circ} .\) Find the lengths of the diagonals.

37–42 Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. \(\cot \theta, \quad \sin \theta ; \quad \theta\) in quadrant II

\(37-42\) me Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 361^{\circ} $$

Two straight roads diverge at an angle of \(65^{\circ} .\) Two cars leave the intersection at \(2 : 00\) P.M., one traveling at 50 \(\mathrm{mi} / \mathrm{h}\) and the other at 30 \(\mathrm{mi} / \mathrm{h} .\) How far apart are the cars at \(2 : 30 \mathrm{P.M.?}\)

37–42 Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. \(\cos \theta, \quad \sin \theta ; \quad \theta\) in quadrant IV

\(37-42\) me Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 1110^{\circ} $$

A car travels along a straight road, heading east for 1 h, then traveling for 30 min on another road that leads northeast. If the car has maintained a constant speed of 40 mi/h, how far is it from its starting position?

37–42 Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. \(\sec \theta, \quad \sin \theta ; \quad \theta\) in quadrant \(\mathrm{I}\)

\(37-42\) me Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -100^{\circ} $$

A pilot flies in a straight path for 1 h 30 min. She then makes a course correction, heading \(10^{\circ}\) to the right of her original course, and flies 2 \(\mathrm{h}\) in the new direction. If she maintains a constant speed of \(625 \mathrm{mi} / \mathrm{h},\) how far is she from her starting position?

37–42 Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. \(\sec \theta, \quad \tan \theta ; \quad \theta\) in quadrant II

\(37-42\) me Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -800^{\circ} $$

Two boats leave the same port at the same time. One travels at a speed of 30 mi/h in the direction N \(50^{\circ} \mathrm{E}\) and the other travels at a speed of 26 \(\mathrm{mi} / \mathrm{h}\) in a direction \(\mathrm{S} 70^{\circ} \mathrm{E}\) (see the figure). How far apart are the two boats after one hour?

37–42 Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. \(\csc \theta, \quad \cot \theta ; \quad \theta\) in quadrant III

\(37-42\) me Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 1270^{\circ} $$

A fisherman leaves his home port and heads in the direction \(\mathrm{N} 70^{\circ} \mathrm{W}\) . He travels 30 \(\mathrm{mi}\) and reaches Egg Island. The next day he sails \(\mathrm{N} 10^{\circ} \mathrm{E}\) for 50 \(\mathrm{mi}\) , reaching Forrest Island. (a) Find the distance between the fisherman’s home port and Forrest Island. (b) Find the bearing from Forrest Island back to his home port.

Number of Solutions in the Ambiguous Case We have seen that when using the Law of Sines to solve a triangle in the SSA case, there may be two, one, or no solution(s). Sketch triangles like those in Figure 6 to verify the criteria in the table for the number of solutions if you are given \(\angle A\) and sides \(a\) and \(b\) . $$ \begin{array}{c|c}{\text { Criterion }} & {\text { Number of Solutions }} \\\ \hline a \geq b & {1} \\ {b>a>b \sin A} & {2} \\ {a=b \sin A} & {1} \\ {a