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CHAPTER 4 REVIEW EXERCISES

CONCEPT CHECK EXERCISES

Determine each of the following as being either true or false.

A standard position angle of $205 °$ $205 °$ is a second-quadrant angle.
In Example 2 of Section 4.2, if the terminal side passes through (6, 8) the values of the trigonometric functions are the same as those shown.
$\csc θ$ $\csc θ$ and $\sin^{- 1} θ$ $\sin^{- 1} θ$ are the same.
In a right triangle with sides a, b, and c, and the standard angles opposite these sides, if $A = 45 °,$ $A = 45 °,$ then $a = b .$ $a = b .$
In Example 2 of Section 4.5, if the angle of elevation from the north goal line to the blimp is $58.5 °$ $58.5 °$ , the solution is the same.
If an angle has a cosine of 0.2, then the secant of the angle is 5.

PRACTICE AND APPLICATIONS

In Exercises 7–10, find the smallest positive angle and the smallest negative angle (numerically) coterminal with but not equal to the given angle.

$47.0 °$ $47.0 °$
$338.8 °$ $338.8 °$
$- 217.5 °$ $- 217.5 °$
$- 0.72 °$ $- 0.72 °$

In Exercises 11–14, express the given angles in decimal form.

$31 ° 5 4^{'}$ $31 ° 5 4^{'}$
$574 ° 4 5^{'}$ $574 ° 4 5^{'}$
$- 83 ° 2 1^{'}$ $- 83 ° 2 1^{'}$
$321 ° 2 7^{'}$ $321 ° 2 7^{'}$

In Exercises 15–18, express the given angles to the nearest minute.

$17.5 °$ $17.5 °$
$- 65.4 °$ $- 65.4 °$
$749.75 °$ $749.75 °$
$126.05 °$ $126.05 °$

In Exercises 19–22, determine the trigonometric functions of the angles (in standard position) whose terminal side passes through the given points. Give answers in exact form.

(24, 7)
(5, 4)
(48, 48)
(0.36, 0.77)

In Exercises 23–28, find the indicated trigonometric functions. Give answers in decimal form, rounded off to three significant digits. Assume $θ$ $θ$ is an acute angle.

Given $\sin θ = \frac{5}{13},$ $\sin θ = \frac{5}{13},$ find $\cos θ$ $\cos θ$ and $\cot θ .$ $\cot θ .$
Given $\cos θ = \frac{3}{8},$ $\cos θ = \frac{3}{8},$ find $\sin θ$ $\sin θ$ and $\tan θ .$ $\tan θ .$
Given $\tan θ = 2,$ $\tan θ = 2,$ find $\cos θ$ $\cos θ$ and $\csc θ .$ $\csc θ .$
Given $\cot θ = 40,$ $\cot θ = 40,$ find $\sin θ$ $\sin θ$ and $\sec θ .$ $\sec θ .$
Given $\cos θ = 0.327,$ $\cos θ = 0.327,$ find $\sin θ$ $\sin θ$ and $\csc θ .$ $\csc θ .$
Given $\csc θ = 3.41,$ $\csc θ = 3.41,$ find $\tan θ$ $\tan θ$ and $\sec θ .$ $\sec θ .$

In Exercises 29–36, find the values of the trigonometric functions. Round off results.

$\sin 72.1 °$ $\sin 72.1 °$
$\cos 40.3 °$ $\cos 40.3 °$
$\tan 85.68 °$ $\tan 85.68 °$
$\sin 0.91 °$ $\sin 0.91 °$
$\sec 36.2 °$ $\sec 36.2 °$
$\csc 82.4 °$ $\csc 82.4 °$
$(\cot 7.06 °) (\sin 7.06 °) - \cos 7.06 °$ $(\cot 7.06 °) (\sin 7.06 °) - \cos 7.06 °$
$(\sec 79.36 °) (\sin 79.36 °) - \tan 79.36 °$ $(\sec 79.36 °) (\sin 79.36 °) - \tan 79.36 °$

In Exercises 37–48, find $θ$ $θ$ for each of the given trigonometric functions. Assume $θ$ $θ$ is an acute angle. Round off results.

$\cos θ = 0.850$ $\cos θ = 0.850$
$\sin θ = 0.63052$ $\sin θ = 0.63052$
$\tan θ = 1.574$ $\tan θ = 1.574$
$\cos θ = 0.0135$ $\cos θ = 0.0135$
$\csc θ = 4.713$ $\csc θ = 4.713$
$\cot θ = 0.7561$ $\cot θ = 0.7561$
$\sec θ = 34.2$ $\sec θ = 34.2$
$\csc θ = 1.92$ $\csc θ = 1.92$
$\cot θ = 7.117$ $\cot θ = 7.117$
$\sec θ = 1.006$ $\sec θ = 1.006$
$\sin θ = 1.030$ $\sin θ = 1.030$
$\tan θ = 0.0052$ $\tan θ = 0.0052$

In Exercises 49 and 50, assume $θ$ $θ$ is an acute angle with the given trigonometric function value. Find the exact coordinates of the point where the terminal side of $θ$ $θ$ (in standard position) intersects the unit circle.

$\sin θ = \frac{2}{5}$ $\sin θ = \frac{2}{5}$
$\tan θ = 3$ $\tan θ = 3$

In Exercises 51–60, solve the right triangles with the given parts. Refer to Fig. 4.68.

Right triangle Ay B C with sides of lower ay, lower b, and lower c opposite angles upper Ay, upper B, and upper C, respectively. — Fig. 4.68

$A = 17.0 °, b = 6.00$ $A = 17.0 °, b = 6.00$
$B = 68.1 °, a = 1080$ $B = 68.1 °, a = 1080$
$a = 81.0, b = 64.5$ $a = 81.0, b = 64.5$
$a = 106, c = 382$ $a = 106, c = 382$
$A = 37.5 °, a = 12.0$ $A = 37.5 °, a = 12.0$
$B = 85.7 °, b = 852.44$ $B = 85.7 °, b = 852.44$
$b = 6.508, c = 7.642$ $b = 6.508, c = 7.642$
$a = 0.721, b = 0.144$ $a = 0.721, b = 0.144$
$A = 49.67 °, c = 0.8253$ $A = 49.67 °, c = 0.8253$
$B = 4.38 °, b = 5682$ $B = 4.38 °, b = 5682$

In Exercises 61–105, solve the given problems.

Find the value of x for the triangle shown in Fig. 4.69.

Fig. 4.69
Explain three ways in which the value of x can be found for the triangle shown in Fig. 4.70. Which of these methods is the easiest?

Fig. 4.70
Find the perimeter of a regular octagon (eight equal sides with equal interior angles) that is inscribed in a circle (all vertices of the octagon touch the circle) of radius 10.
Explain why values of $\sin θ$ $\sin θ$ increase as $θ$ $θ$ increases from $0 °$ $0 °$ to $90 °$ $90 °$ .
What is x if (3, 2) and (x, 7) are on the same terminal side of an acute angle?
Two legs of a right triangle are 2.607 and 4.517. What is the smaller acute angle?
Show that the side c of any triangle ABC is related to the perpendicular h from C to side AB by the equation

$c = h \cot A + h \cot B .$ $c = h \cot A + h \cot B .$
For the isosceles triangle shown in Fig. 4.71, show that $c = 2 a \sin \frac{A}{2} .$ $c = 2 a \sin \frac{A}{2} .$

Fig. 4.71
In Fig. 4.72, find the length c of the chord in terms of r and the angle $θ / 2.$ $θ / 2.$

Fig. 4.72
In Fig. 4.73, find a formula for h in terms of d, $α,$ $α,$ and $β .$ $β .$

Fig. 4.73
Find the angle between the line passing through the origin and (3, 2), and the line passing through the origin and (2, 3).
A sloped cathedral ceiling is between walls that are 7.50 ft high and 12.0 ft high. If the walls are 15.0 ft apart, at what angle does the ceiling rise?
The base of a 75-ft fire truck ladder is at the rear of the truck and is 4.8 ft above the ground. If the ladder is extended backward at an angle of $62 °$ $62 °$ with the horizontal, how high up on the building does it reach, and how far from the building must the back of the truck be in order that the ladder just reach the building? See Fig. 4.74.

Fig. 4.74

Figure 4.74 Full Alternative Text
A pendulum 1.25 m long swings through an angle of $5.6 °$ $5.6 °$ . What is the distance between the extreme positions of the pendulum?
The voltage e at any instant in a coil of wire that is turning in a magnetic field is given by $e = E \cos α,$ $e = E \cos α,$ where E is the maximum voltage and $α$ $α$ is the angle the coil makes with the field. Find the acute angle $α$ $α$ if $e = 56.9 V$ $e = 56.9 V$ and $E = 339 V .$ $E = 339 V .$
The area of a quadrilateral with diagonals $d_{1}$ $d_{1}$ and $d_{2}$ $d_{2}$ is $A = \frac{1}{2} d_{1} d_{2} \sin θ,$ $A = \frac{1}{2} d_{1} d_{2} \sin θ,$ where $d_{1}$ $d_{1}$ and $d_{2}$ $d_{2}$ are the diagonals and $θ$ $θ$ is the angle between them. Find the area of an approximately four-sided grass fire with diagonals of 320 ft and 440 ft and $θ = 72 ° .$ $θ = 72 ° .$
For a car rounding a curve, the road should be banked at an angle $θ$ $θ$ according to the equation $\tan θ = \frac{v^{2}}{g r} .$ $\tan θ = \frac{v^{2}}{g r} .$ Here, v is the speed of the car and r is the radius of the curve in the road. See Fig. 4.75. Find $θ$ $θ$ for $v = 80.7 ft / s (55.0 mi / h), g = 32.2 ft / s^{2},$ $v = 80.7 ft / s (55.0 mi / h), g = 32.2 ft / s^{2},$ and $r = 950 ft .$ $r = 950 ft .$

Fig. 4.75
The apparent power S in an electric circuit in which the power is P and the impedance phase angle is $θ$ $θ$ is given by $S = P \sec θ .$ $S = P \sec θ .$ Given $P = 12.0 V \cdot A$ $P = 12.0 V \cdot A$ and $θ = 29.4 °,$ $θ = 29.4 °,$ find S.
A surveyor measures two sides and the included angle of a triangular tract of land to be $a = 31.96 m, b = 47.25 m,$ $a = 31.96 m, b = 47.25 m,$ and $C = 64.09 ° .$ $C = 64.09 ° .$ (a) Show that a formula for the area A of the tract is $A = \frac{1}{2} a b \sin C .$ $A = \frac{1}{2} a b \sin C .$ (b) Find the area of the tract.
A water channel has the cross section of an isosceles trapezoid. See Fig. 4.76. (a) Show that a formula for the area of the cross section is $A = b h + h^{2} \cot θ .$ $A = b h + h^{2} \cot θ .$ (b) Find A if $b = 12.6 ft, h = 4.75 ft,$ $b = 12.6 ft, h = 4.75 ft,$ and $θ = 37.2 ° .$ $θ = 37.2 ° .$

Fig. 4.76
In tracking an airplane on radar, it is found that the plane is 27.5 km on a direct line from the control tower, with an angle of elevation of $10.3 °$ $10.3 °$ . What is the altitude of the plane?
A straight emergency chute for an airplane is 16.0 ft long. In being tested, the top of the chute is 8.5 ft above the ground. What angle does the chute make with the ground?
The windshield on an automobile is inclined $42.5 °$ $42.5 °$ with respect to the horizontal. Assuming that the windshield is flat and rectangular, what is its area if it is 4.80 ft wide and the bottom is 1.50 ft in front of the top?
A water slide at an amusement park is 85 ft long and is inclined at an angle of $52 °$ $52 °$ with the horizontal. How high is the top of the slide above the water level?
Find the area of the patio shown in Fig. 4.77.

Fig. 4.77

Figure 4.77 Full Alternative Text
The cross section (a regular trapezoid) of a levee to be built along a river is shown in Fig. 4.78. What is the volume of rock and soil that will be needed for a one-mile length of the levee?

Fig. 4.78
The vertical cross section of an attic room in a house is shown in Fig. 4.79. Find the distance d across the floor.

Fig. 4.79
The impedance Z and resistance R in an AC circuit may be represented by letting the impedance be the hypotenuse of a right triangle and the resistance be the side adjacent to the phase angle $ϕ .$ $ϕ .$ If $R = 1.75 \times 10^{3} Ω$ $R = 1.75 \times 10^{3} Ω$ and $ϕ = 17.38 °,$ $ϕ = 17.38 °,$ find Z.
A typical aqueduct built by the Romans dropped on average at an angle of about $0.03 °$ $0.03 °$ to allow gravity to move the water from the source to the city. For such an aqueduct of 65 km in length, how much higher was the source than the city?
The distance from the ground level to the underside of a cloud is called the ceiling. See Fig. 4.80. A ground observer 950 m from a searchlight aimed vertically notes that the angle of elevation of the spot of light on a cloud is $76 °$ $76 °$ . What is the ceiling?

Fig. 4.80
The window of a house is shaded as shown in Fig. 4.81. What percent of the window is shaded when the angle of elevation $θ$ $θ$ of the sun is $65 °$ $65 °$ ?

Fig. 4.81

Figure 4.81 Full Alternative Text
A person standing on a level plain hears the sound of a plane, looks in the direction of the sound, but the plane is not there (familiar?). When the sound was heard, it was coming from a point at an angle of elevation of $25 °$ $25 °$ , and the plane was traveling at 450 mi/h (660 ft/s) at a constant altitude of 2800 ft along a straight line. If the plane later passes directly over the person, at what angle of elevation should the person have looked directly to see the plane when the sound was heard? (The speed of sound is 1130 ft/s.) See Fig. 4.82.

Fig. 4.82
In the structural support shown in Fig. 4.83, find x.

Fig. 4.83

Figure 4.83 Full Alternative Text
The main span of the Mackinac Bridge (see Fig. 4.84) in northern Michigan is 1160 m long. The angle subtended by the span at the eye of an observer in a helicopter is $2.2 °$ $2.2 °$ . Show that the distance calculated from the helicopter to the span is about the same if the line of sight is perpendicular to the end or to the middle of the span.

Fig. 4.84
A Coast Guard boat 2.75 km from a straight beach can travel at 37.5 km/h. By traveling along a line that is at $69.0 °$ $69.0 °$ with the beach, how long will it take it to reach the beach? See Fig. 4.85.

Fig. 4.85
Each side piece of the trellis shown in Fig. 4.86 makes an angle of $80.0 °$ $80.0 °$ with the ground. Find the length of each side piece and the area covered by the trellis.

Fig. 4.86
A laser beam is transmitted with a “width” of $0.00200 °$ $0.00200 °$ . What is the diameter of a spot of the beam on an object 52,500 km distant? See Fig. 4.87.

Fig. 4.87
The surface of a soccer ball consists of 20 regular hexagons (six sides) interlocked around 12 regular pentagons (five sides). See Fig. 4.88. (a) If the side of each hexagon and pentagon is 45.0 mm, what is the surface area of the soccer ball? (b) Find the surface area, given that the diameter of the ball is 222 mm, (c) Assuming that the given values are accurate, account for the difference in the values found in parts (a) and (b).

Fig. 4.88
Through what angle $θ$ $θ$ must the crate shown in Fig. 4.89 be tipped in order that its center of gravity C is directly above the pivot point P?

Fig. 4.89
Find the gear angle $θ$ $θ$ in Fig. 4.90, if $t = 0.180 in .$ $t = 0.180 in .$

Fig. 4.90
A hang glider is directly above the shore of a lake. An observer on a hill is 375 m along a straight line from the shore. From the observer, the angle of elevation of the hang glider is $42.0 °$ $42.0 °$ , and the angle of depression of the shore is $25.0 °$ $25.0 °$ . How far above the shore is the hang glider?
A ground observer sights a weather balloon to the east at an angle of elevation of $15.0 °$ $15.0 °$ . A second observer 2.35 mi to the east of the first also sights the balloon to the east at an angle of elevation of $24.0 °$ $24.0 °$ . How high is the balloon? See Fig. 4.91.

Fig. 4.91
A uniform strip of wood 5.0 cm wide frames a trapezoidal window, as shown in Fig. 4.92. Find the left dimension l of the outside of the frame.

Fig. 4.92
A crop-dusting plane flies over a level field at a height of 25 ft. If the dust leaves the plane through a $30 °$ $30 °$ angle and hits the ground after the plane travels 75 ft, how wide a strip is dusted? See Fig. 4.93.

Fig. 4.93
A patio is designed in the shape of an isosceles trapezoid with bases 5.0 m and 7.0 m. The other sides are 6.0 m each. Write one or two paragraphs explaining how to use (a) the sine and (b) the cosine to find the internal angles of the patio, and (c) the tangent in finding the area of the patio.