Chapter 6

Prove that in triangle \(A B C\) $$\begin{array}{l}{a=b \cos C+c \cos B} \\ {b=c \cos A+a \cos C} \\ {c=a \cos B+b \cos A}\end{array}$$ 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.]

Find the quadrant in which \(\theta\) lies from the information given. $$ \csc \theta>0 \quad \text { and } \quad \cos \theta<0 $$

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ 50^{\circ}, \quad 340^{\circ} $$

Height of the Space Shuttle An observer views the space shuttle from a distance of 2 \(\mathrm{mi}\) from the launch pad. (a) Express the height of the space shuttle as a function of the angle of elevation \(\theta\) . (b) Express the angle of elevation \(\theta\) as a function of the height \(h\) of the space shuttle.

Height of a Tree A tree on a hillside casts a shadow 215 \(\mathrm{ft}\) 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.

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$ \tan \theta, \quad \cos \theta ; \quad \theta \text { in Quadrant III } $$

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.

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$ \cot \theta, \quad \sin \theta ; \quad \theta \text { in Quadrant II } $$

Using a protractor, sketch a right triangle that has the acute angle \(40^{\circ} .\) Measure the sides carefully, and use your results to estimate the six trigonometric ratios of \(40^{\circ} .\)

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 \(2 : 30 \mathrm{P.M.?}\)

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$ \cos \theta, \quad \sin \theta ; \quad \theta \text { in Quadrant IV } $$

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?

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

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

Surfing the Perfect Wave For a wave to be surfable, it can't break all at once. Robert Guza and Tony Bowen have shown that a wave has a surfable shoulder if it hits the shoreline at an angle \(\theta\) given by $$ \theta=\sin ^{-1}\left(\frac{1}{(2 n+1) \tan \beta}\right) $$ where \(\beta\) is the angle at which the beach slopes down and where \(n=0,1,2, \ldots\) (a) For \(\beta=10^{\circ},\) find \(\theta\) when \(n=3\) (b) For \(\beta=15^{\circ},\) find \(\theta\) when \(n=2,3,\) and \(4 .\) Explain why the formula does not give a value for \(\theta\) when \(n=0\) or 1

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$ \sec \theta, \quad \tan \theta ; \quad \theta \text { in Quadrant II } $$

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

Inverse Trigonometric Functions on a Calculator Most calculators do not have keys for sec \(^{-1}, \mathrm{csc}^{-1},\) or cot \(^{-1}\) . Prove the following identities, and then use these identities and a calculator to find \(\sec ^{-1} 2, \mathrm{csc}^{-1} 3,\) and \(\cot ^{-1} 4 .\) $$\begin{array}{ll}{\sec ^{-1} x=\cos ^{-1}\left(\frac{1}{x}\right),} & {x \geq 1} \\ {\csc ^{-1} x=\sin ^{-1}\left(\frac{1}{x}\right),} & {x \geq 1} \\\ {\cot ^{-1} x=\tan ^{-1}\left(\frac{1}{x}\right),} & {x>0}\end{array}$$

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$ \csc \theta, \quad \cot \theta ; \quad \theta \text { in Quadrant III } $$

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.

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \sin \theta=\frac{3}{5}, \quad \theta \text { in Quadrant II } $$

Find an angle between 0 and 2p that is coterminal with the given angle. $$ \frac{17 \pi}{6} $$

Airport B is 300 mi from airport A at a bearing \(\mathrm{N} 50^{\circ} \mathrm{E}\) (see the figure). A pilot wishing to fly from \(\mathrm{A}\) to \(\mathrm{B}\) mistakenly flies due east at 200 \(\mathrm{mi} / \mathrm{h}\) for 30 minutes, when he notices his error. (a) How far is the pilot from his destination at the time he notices the error? (b) What bearing should he head his plane in order to arrive at airport B?

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \cos \theta=-\frac{7}{12}, \quad \theta \text { in Quadrant III } $$

Find an angle between 0 and 2p that is coterminal with the given angle. $$ -\frac{7 \pi}{3} $$

A triangular field has sides of lengths 22, 36, and 44 yd. Find the largest angle.

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \tan \theta=-\frac{3}{4}, \quad \cos \theta>0 $$

Height of a Building The angle of elevation to the top of the Empire State Building in New York is found to be \(11^{\circ}\) from the ground at a distance of 1 \(\mathrm{mi}\) from the base of the building. Using this information, find the height of the Empire State Building.

Find an angle between 0 and 2p that is coterminal with the given angle. $$87 \pi$$

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \sec \theta=5, \quad \sin \theta<0 $$

Gateway Arch A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of \(35,000\) ft. The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is \(22^{\circ} .\) (a) What is the distance between the plane and the arch? (b) What is the distance between a point on the ground directly below the plane and the arch?

Find an angle between 0 and 2p that is coterminal with the given angle. $$ 10 $$

A boy is flying two kites at the same time. He has 380 ft of line out to one kite and 420 ft to the other. He estimates the angle between the two lines to be \(30^{\circ} .\) Approximate the distance between the kites.

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \csc \theta=2, \quad \theta \text { in Quadrant } \mathrm{I} $$

Deviation of a Laser Beam A laser beam is to be directed toward the center of the moon, but the beam strays \(0.5^{\circ}\) from its intended path. (a) How far has the beam diverged from its assigned target when it reaches the moon? (The distance from the earth to the moon is \(240,000\) mi. (b) The radius of the moon is about 1000 mi. Will the beam strike the moon?

Find an angle between 0 and 2p that is coterminal with the given angle. $$ \frac{17 \pi}{4} $$

A 125-ft tower is located on the side of a mountain that is inclined \(32^{\circ}\) to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 \(\mathrm{ft}\) downhill from the base of the tower. Find the shortest length of wire needed.

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \cot \theta=\frac{1}{4}, \quad \sin \theta<0 $$

Distance at Sea From the top of a 200 -ft lighthouse, the angle of depression to a ship in the ocean is \(23^{\circ} .\) How far is the ship from the base of the lighthouse?

Find an angle between 0 and 2p that is coterminal with the given angle. $$ \frac{51 \pi}{2} $$

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \cos \theta=-\frac{2}{7}, \quad \tan \theta<0 $$

Leaning Ladder \(A 20-f t\) ladder leans against a building so that the angle between the ground and the ladder is \(72^{\circ} .\) How high does the ladder reach on the building?

The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, 1150 ft above the ground, wants to determine the distance between two landmarks on the ground below. She observes that the angle formed by the lines of sight to these two landmarks is \(43^{\circ} .\) She also observes that the angle between the vertical and the line of sight to one of the landmarks is \(62^{\circ}\) and to the other landmark is \(54^{\circ} .\) Find the distance between the two landmarks.

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \tan \theta=-4, \quad \sin \theta>0 $$

Height of a Tower \(A 600-f t\) guy wire is attached to the top of a communications tower. If the wire makes an angle of \(65^{\circ}\) with the ground, how tall is the communications tower?

Land in downtown Columbia is valued at \(\$ 20\) a square foot. What is the value of a triangular lot with sides of lengths 112, 148, and 190 ft?