Chapter 6

If \(\theta=\pi / 3,\) find the value of each expression. \(\begin{array}{ll}{\text { (a) } \sin 2 \theta,} & {2 \sin \theta} & {\text { (b) } \sin \frac{1}{2} \theta, \quad \frac{1}{2} \sin \theta} \\ {\text { (c) } \sin ^{2} \theta,} & {\sin \left(\theta^{2}\right)}\end{array}\)

Elevation of a Kite \(\quad\) A man is lying on the beach, flying a kite. He holds the end of the kite string at ground level, and estimates the angle of elevation of the kite to be \(50^{\circ} .\) If the string is 450 \(\mathrm{ft}\) long, how high high is the kite above the ground?

Find the area of a triangle with sides of length 7 and 9 and included angle \(72^{\circ} .\)

Determining a Distance A woman standing on a hill sees a flagpole that she knows is 60 \(\mathrm{ft}\) tall. The angle of depression to the bottom of the pole is \(14^{\circ},\) and the angle of elevation to the top of the pole is \(18^{\circ} .\) Find her distance \(x\) from the pole.

Find the length of an arc that subtends a central angle of \(45^{\circ}\) in a circle of radius \(10 \mathrm{m} .\)

Find the area of a triangle with sides of length 10 and 22 and included angle \(10^{\circ} .\)

Height of a Tower \(A\) water tower is located 325 \(\mathrm{ft}\) from a building (see the figure). From a window in the building, an observer notes that the angle of elevation to the top of the tower is \(39^{\circ}\) and that the angle of depression to the bottom of the tower is \(25^{\circ} .\) How tall is the tower? How high is the window?

Find the length of an arc that subtends a central angle of 2 rad in a circle of radius \(2 \mathrm{mi} .\)

Find the area of an equilateral triangle with side of length \(10 .\)

A central angle \(\theta\) in a circle of radius 5 \(\mathrm{m}\) is subtended by an arc of length \(6 \mathrm{m} .\) Find the measure of \(\theta\) in degrees and in radians.

Determining a Distance An airplane is flying at an elevation of 5150 ft, directly above a straight highway. Two motorists are driving cars on the highway on opposite sides of the plane, and the angle of depression to one car is \(35^{\circ}\) and to the other is \(52^{\circ} .\) How far apart are the cars?

A triangle has area of \(16 \mathrm{in}^{2},\) and two of the sides of the triangle have lengths 5 in. and 7 in. Find the angle included by these two sides.

An arc of length 100 \(\mathrm{m}\) subtends a central angle \(\theta\) in a circle of radius 50 \(\mathrm{m}\) . Find the measure of \(\theta\) in degrees and in radians.

An isosceles triangle has an area of \(24 \mathrm{cm}^{2},\) and the angle between the two equal sides is \(5 \pi / 6 .\) What is the length of the two equal sides?

A circular arc of length 3 \(\mathrm{ft}\) subtends a central angle of \(25^{\circ}\) . Find the radius of the circle.

Height of a Balloon A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be \(20^{\circ}\) and \(22^{\circ}\) . How high is the balloon?

Find the radius of the circle if an arc of length 6 \(\mathrm{m}\) on the circle subtends a central angle of \(\pi / 6 \mathrm{rad}\) .

Height of a Mountain To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be \(32^{\circ} .\) One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is \(35^{\circ} .\) Estimate the height of the mountain.

Find the radius of the circle if an arc of length 4 \(\mathrm{ft}\) on the circle subtends a central angle of \(135^{\circ} .\)

Height of Cloud Cover To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle \(75^{\circ}\) from the horizontal. An observer 600 \(\mathrm{m}\) away measures the angle of elevation to the spot of light to be \(45^{\circ} .\) Find the height \(h\) of the cloud cover.

Use the first Pythagorean identity to prove the second. [Hint: Divide by \(\cos ^{2} \theta .\)]

Distance to the Sun When the moon is exactly half full, the earth, moon, and sun form a right angle (see the figure). At that time the angle formed by the sun, earth, and moon is measured to be \(89.85^{\circ} .\) If the distance from the earth to the moon is \(240,000 \mathrm{mi}\) , estimate the distance from the earth to the sun.

Height of a Rocket A rocket fired straight up is tracked by an observer on the ground a mile away. (a) Show that when the angle of elevation is \(\theta\) , the height of the rocket in feet is \(h=5280 \tan \theta\) . (b) Complete the table to find the height of the rocket at the given angles of elevation. $$ \begin{array}{|c|c|c|c|c|}\hline \theta & {20^{\circ}} & {60^{\circ}} & {80^{\circ}} & {85^{\circ}} \\ \hline h & {} & {} & {} \\ \hline\end{array} $$

Find the area of a sector with central angle 1 rad in a circle of radius \(10 \mathrm{m} .\)

Rain Gutter A rain gutter is to be constructed from a metal sheet of width 30 \(\mathrm{cm}\) by bending up one-third of the sheet on each side through an angle \(\theta .\) (a) Show that the cross-sectional area of the gutter is modeled by the function $$ A(\theta)=100 \sin \theta+100 \sin \theta \cos \theta $$ (b) Graph the function \(A\) for \(0 \leq \theta \leq \pi / 2\) (c) For what angle \(\theta\) is the largest cross-sectional area achieved?

A sector of a circle has a central angle of \(60^{\circ} .\) Find the area of the sector if the radius of the circle is \(3 \mathrm{mi} .\)

Parallax To find the distance to nearby stars, the method of parallax is used. The idea is to find a triangle with the star at one vertex and with a base as large as possible. To do this, the star is observed at two different times exactly 6 months apart, and its apparent change in position is recorded. From these two observations, \(\angle E_{1} S E_{2}\) can be calculated. The times are chosen so that \(\angle E_{1} S E_{2}\) is as large as possible, which guarantees that \(\angle E_{1}\) OS is \(90^{\circ} .\) The angle \(E_{1} S O\) is called the parallax of the star. Alpha Centauri, the star nearest the earth, has a parallax of \(0.000211^{\circ} .\) Estimate the distance to this star. (Take the distance from the earth to the sun to be \(9.3 \times 10^{7} \mathrm{mi} .\) ).

The area of a sector of a circle with a central angle of 2 rad is \(16 \mathrm{m}^{2} .\) Find the radius of the circle.

Distance from Venus to the Sun The elongation \(\alpha\) of a planet is the angle formed by the planet, earth, and sun (see the figure). When Venus achieves its maximum elongation of \(46.3^{\circ},\) the earth, Venus, and the sun form a triangle with a right angle at Venus. Find the distance between Venus and the sun in astronomical units (AU). (By definition the distance between the earth and the sun is 1 AU.)

A sector of a circle of radius 24 \(\mathrm{mi}\) has an area of 288 \(\mathrm{mi}^{2}\) . Find the central angle of the sector.

Similar Triangles If two triangles are similar, what properties do they share? Explain how these properties make it possible to define the trigonometric ratios without regard to the size of the triangle.

Throwing a Shot Put The range \(R\) and height \(H\) of a shot put thrown with an initial velocity of \(v_{0} \mathrm{ft} / \mathrm{s}\) at an angle \(\theta\) are given by $$ \begin{array}{l}{R=\frac{v_{0}^{2} \sin (2 \theta)}{g}} \\ {H=\frac{v_{0}^{2} \sin ^{2} \theta}{2 g}}\end{array} $$ On the earth \(g=32 \mathrm{ft} / \mathrm{s}^{2}\) and on the moon \(g=5.2 \mathrm{ft} / \mathrm{s}^{2} .\) Find the range and height of a shot put thrown under the given conditions. (a) On the earth with \(v_{0}=12 \mathrm{ft} / \mathrm{s}\) and \(\theta=\pi / 6\) (b) On the moon with \(v_{0}=12 \mathrm{ft} / \mathrm{s}\) and \(\theta=\pi / 6\)

The area of a circle is \(72 \mathrm{cm}^{2} .\) Find the area of a sector of this circle that subtends a central angle of \(\pi / 6 \mathrm{rad}\) .

Sledding The time in seconds that it takes for a sled to slide down a hillside inclined at an angle \(\theta\) is $$ t=\sqrt{\frac{d}{16 \sin \theta}} $$ where \(d\) is the length of the slope in feet. Find the time it takes to slide down a \(2000-\mathrm{ft}\) slope inclined at \(30^{\circ} .\)

Beehives In a beehive each cell is a regular hexagonal prism, as shown in the figure. The amount of wax \(W\) in the cell depends on the apex angle \(\theta\) and is given by $$ W=3.02-0.38 \cot \theta+0.65 \csc \theta $$ Bees instinctively choose \(\theta\) so as to use the least amount of wax possible. (a) Use a graphing device to graph Was a function of \(\theta\) for \(0 < \theta < \pi\) (b) For what value of \(\theta\) does \(W\) have its minimum value? [Note: Biologists have discovered that bees rarely deviate from this value by more than a degree or two. \(]\)

Travel Distance A car's wheels are 28 in. in diameter. How far (in miles) will the car travel if its wheels revolve \(10,000\) times without slipping?

Turning a Corner A steel pipe is being carried down a hallway that is 9 \(\mathrm{ft}\) wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 \(\mathrm{ft}\) wide. (a) Show that the length of the pipe in the figure is modeled by the function $$ L(\theta)=9 \csc \theta+6 \sec \theta $$ (b) Graph the function \(L\) for \(0<\theta<\pi / 2\) (c) Find the minimum value of the function \(L\) (d) Explain why the value of \(L\) you found in part (c) is the length of the longest pipe that can be carried around the corner.

Wheel Revolutions How many revolutions will a car wheel of diameter 30 in. make as the car travels a distance of one mile?

Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation \(\theta\) of a rainbow is always the same. It can be shown that \(\theta=4 \beta-2 \alpha,\) where $$ \sin \alpha=k \sin \beta $$ and \(\alpha=59.4^{\circ}\) and \(k=1.33\) is the index of refraction of water. Use the given information to find the angle of elevation \(\theta\) of a rainbow. (For a mathematical explanation of rainbows see Calculus Early Transcendentals, 7 th Edition, by James Stewart, page 282 .)

Latitudes Pittsburgh, Pennsylvania, and Miami, Florida, lie approximately on the same meridian. Pittsburgh has a latitude of \(40.5^{\circ} \mathrm{N},\) and Miami has a latitude of \(25.5^{\circ} \mathrm{N} .\) Find the distance between these two cities. (The radius of the earth is 3960 \(\mathrm{mi}\) .)

Using a Calculator To solve a certain problem, you need to find the sine of 4 rad. Your study partner uses his calculator and tells you that $$ \sin 4=0.0697564737 $$ On your calculator you get $$ \sin 4=-0.7568024953 $$ What is wrong? What mistake did your partner make?

Latitudes Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has a latitude of \(35^{\circ} \mathrm{N}\) , and New Orleans has a latitude of \(30^{\circ} \mathrm{N}\) . Find the distance between these two cities. (The radius of the earth is 3960 \(\mathrm{mi}\) .)

Orbit of the Earth Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles. IThe path of the earth around the sun is actually an ellipse with the sun at one focus (see Section \(11.2 ) .\) This ellipse, however, has very small eccentricity, so it is nearly circular.]

Nautical Miles Find the distance along an arc on the sur- face of the earth that subtends a central angle of 1 minute \(\left(1 \text { minute }=\frac{1}{60} \text { degree). This distance is called a nautical mile. }\right.\) (The radius of the earth is 3960 \(\mathrm{mi}\) .)

Irrigation An irrigation system uses a straight sprinkler pipe 300 \(\mathrm{ft}\) long that pivots around a central point as shown. Due to an obstacle the pipe is allowed to pivot through \(280^{\circ}\) only. Find the area irrigated by this system.

Windshield Wipers The top and bottom ends of a wind-shield wiper blade are 34 in. and 14 in, respectively, from the pivot point. While in operation, the wiper sweeps through \(135^{\circ} .\) Find the area swept by the blade.

Fan A ceiling fan with 16 -in. blades rotates at 45 \(\mathrm{rpm}\) . (a) Find the angular speed of the fan in rad/min. (b) Find the linear speed of the tips of the blades in in./min.

Radial Saw A radial saw has a blade with a 6 -in. radius. Suppose that the blade spins at 1000 \(\mathrm{rpm}\) . (a) Find the angular speed of the blade in rad/min. (b) Find the linear speed of the sawteeth in \(\mathrm{ft} / \mathrm{s}\) .

Winch A winch of radius 2 \(\mathrm{ft}\) is used to lift heavy loads. If the winch makes 8 revolutions every 15 \(\mathrm{s}\) , find the speed at which the load is rising.

Speed of a Car The wheels of a car have radius 11 in. and are rotating at \(600 \mathrm{rpm} .\) Find the speed of the car in mi/h.