Answer Key
Precalculus Student Solution Chapter 8
-
University:
Santa Ana College -
Course:
MATH 170 | Pre-Calculus Mathematics Academic year:
2019
-
Views:
339
Pages:
44
Author:
omiliar74be
178
Chapter 8
Try It
8.1.
α = 98∘
β = 39∘
γ = 43∘
a = 34.6
b = 22
c = 23.8
8.2. Solution 1
(8.16)
α = 80° a = 120
β ≈ 83.2° b = 121
γ ≈ 16.8° c ≈ 35.2
Solution 2
(8.17) 179
α′ = 80°
β′ ≈ 96.8°
γ′ ≈ 3.2°
8.3.
β ≈ 5.7°, γ ≈ 94.3°, c ≈ 101.3
8.4. two
8.5. about
8.2 square feet
8.6. 161.9 yd.
8.7.
a ≈ 14.9, β ≈ 23.8°, γ ≈ 126.2°.
8.8.
α ≈ 27.7°, β ≈ 40.5°, γ ≈ 111.8°
8.9. Area = 552 square feet
8.10. about 8.15 square feet
8.11.
8.12.
8.13.
⎛
⎞
(x, y) = ⎝1 , − 3 ⎠
2
2
8.14.
r= 3
a′ = 120
b′ = 121
c′ ≈ 6.8 180
8.15.
x 2 + y 2 = 2y or, in the standard form for a circle,
x 2 + ⎛⎝y − 1⎞⎠ 2 = 1
8.16. The equation fails the symmetry test with respect to the line
θ=π
2 and with respect to the pole. It passes the polar axis symmetry test.
8.17. Tests will reveal symmetry about the polar axis. The zero is
⎛ π⎞
⎝0, 2 ⎠,
(3, 0).
and the maximum value is
8.18.
8.19. The graph is a rose curve,
n even
8.20. 181
Rose curve,
n odd
8.21.
8.22.
8.23. 13
8.24.
|z| = 50 = 5 2
8.25.
⎛
⎛ ⎞
⎛ ⎞⎞
z = 3⎝cos⎝π ⎠ + isin⎝π ⎠⎠
2
2 182
8.26.
⎛ ⎞⎞
⎛ ⎛ ⎞
z = 2⎝cos⎝π ⎠ + isin⎝π ⎠⎠
6
6
8.27.
z = 2 3 − 2i
8.28.
z
z 1 z 2 = − 4 3; z 1 = − 3 + 3 i
2
2
2
8.29.
z 0 = 2(cos(30°) + isin(30°))
z 1 = 2(cos(120°) + isin(120°))
z 2 = 2(cos(210°) + isin(210°))
z 3 = 2(cos(300°) + isin(300°))
8.30.
8.31.
x(t) = t 3 − 2t
y(t) = t
8.32.
y=5−
8.33.
y = ln x
1x − 3
2
t
x(t)
y(t)
−1
−4
2
0
−3
4
1
−2
6
2
−1
8 183
8.34.
2
x2 + y = 1
4
9
8.35.
y = x2
8.36.
8.37.
8.38. The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the
parametric equations. 184
8.39.
8.40.
3u = 〈 15, 12 〉
8.41.
u = 8i − 11 j
8.42.
v = 34cos(59°)i + 34sin(59°) j Magnitude =
34
⎛ ⎞
θ = tan −1 ⎝5 ⎠ = 59.04°
3
Section Exercises
1. The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.
3. When the known values are the side opposite the missing angle and another side and its opposite angle.
5. A triangle with two given sides and a non-included angle.
7.
β = 72°, a ≈ 12.0, b ≈ 19.9
9.
γ = 20°, b ≈ 4.5, c ≈ 1.6
11.
b ≈ 3.78
13.
c ≈ 13.70 185
15. one triangle,
α ≈ 50.3°, β ≈ 16.7°, a ≈ 26.7
17. two triangles,
γ ≈ 54.3°, β ≈ 90.7°, b ≈ 20.9 or
γ′ ≈ 125.7°, β′ ≈ 19.3°, b′ ≈ 6.9
19. two triangles,
β ≈ 75.7°,
γ ≈ 61.3°, b ≈ 9.9 or
β′ ≈ 18.3°, γ′ ≈ 118.7°, b′ ≈ 3.2
21. two triangles,
α ≈ 143.2°, β ≈ 26.8°, a ≈ 17.3 or
α′ ≈ 16.8°, β′ ≈ 153.2°, a′ ≈ 8.3
23. no triangle possible
25.
A ≈ 47.8° or
A′ ≈ 132.2°
27.
8.6
29.
370.9
31.
12.3
33.
12.2
35.
16.0
37.
29.7°
39.
x = 76.9°or x = 103.1°
41.
110.6°
43.
A ≈ 39.4, C ≈ 47.6, BC ≈ 20.7
45.
57.1
47.
42.0
49.
430.2
51.
10.1
53.
AD ≈ 13.8
55.
AB ≈ 2.8
57.
L ≈ 49.7, N ≈ 56.3, LN ≈ 5.8
59. 51.4 feet
61. The distance from the satellite to station
A is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.
63. 2.6 ft
65. 5.6 km
67. 371 ft
69. 5936 ft 186
71. 24.1 ft
73. 19,056 ft2
75. 445,624 square miles
77. 8.65 ft2
78. two sides and the angle opposite the missing side.
80.
s is the semi-perimeter, which is half the perimeter of the triangle.
82. The Law of Cosines must be used for any oblique (non-right) triangle.
84. 11.3
86. 34.7
88. 26.7
90. 257.4
92. not possible
94. 95.5°
96. 26.9°
98.
B ≈ 45.9°, C ≈ 99.1°, a ≈ 6.4
100.
A ≈ 20.6°, B ≈ 38.4°, c ≈ 51.1
102.
A ≈ 37.8°, B ≈ 43.8, C ≈ 98.4°
104. 177.56 in2
106. 0.04 m2
108. 0.91 yd2
110. 3.0
112. 29.1
114. 0.5
116. 70.7°
118. 77.4°
120. 25.0
122. 9.3
124. 43.52
126. 1.41
128. 0.14
130. 18.3
132. 48.98
134.
136. 7.62
138. 85.1
140. 24.0 km
142. 99.9 ft
144. 37.3 miles
146. 2371 miles
148.
150. 599.8 miles
152. 65.4 cm2
154. 468 ft2 187
156. For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while
in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate
plane, there is one representation, but for each point in the polar plane, there are infinite representations.
158. Determine
θ for the point, then move
r units from the pole to plot the point. If
r is negative, move
r units from the pole in the opposite direction but along the same angle. The point is a distance of
r away from the origin at an angle of
θ from the polar axis.
160. The point
⎛
π⎞
⎝−3, 2 ⎠
has a positive angle but a negative radius and is plotted by moving to an angle of
π
2 and then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The point
⎛
π⎞
⎝3, − 2 ⎠
has a negative angle and a positive radius and is plotted by first moving to an angle of
−π
2 and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the
negative y-axis.
162.
(−5, 0)
164.
⎛ 3 3
3⎞
⎝− 2 , − 2 ⎠
166.
⎛
⎝
2 5,
0.464⎞⎠
168.
⎛
⎝
34, 5.253⎞⎠
170.
⎛
π⎞
⎝8 2, 4 ⎠
172.
r = 4cscθ
174.
r=
176.
3
sinθ
2cos 4 θ
r = 3cosθ
178.
r = 3sinθ
cos(2θ)
180.
r = 9sinθ
cos 2 θ
182.
r=
184.
1
9cosθsinθ
x 2 + y 2 = 4x or
(x − 2) 2 y 2
+
= 1;
4
4
circle
186.
3y + x = 6; line 188
188.
y = 3; line
190.
xy = 4; hyperbola
192.
x 2 + y 2 = 4; circle
194.
x − 5y = 3; line
196.
⎛ 3π ⎞
⎝3, 4 ⎠
198.
(5, π)
200.
202.
204. 189
206.
208.
210.
r=
6
5cosθ − sinθ 190
212.
r = 2sinθ
214.
r=
216.
2
cosθ
r = 3cosθ 191
218.
x 2 + y 2 = 16
220.
y=x
222.
x 2 + ⎛⎝y + 5⎞⎠ 2 = 25 192
224.
(1.618, − 1.176)
226.
(10.630, 131.186°)
228.
(2, 3.14)or(2, π)
230. A vertical line with
a units left of the y-axis.
232. A horizontal line with
a units below the x-axis.
234.
236. 193
238.
240. Symmetry with respect to the polar axis is similar to symmetry about the
x -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line
θ=π
2 is similar to symmetry about the
y -axis.
242. Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid,
limaçon, lemniscate, etc., then plot points at
θ = 0, π , π and 3π ,
2
2 and sketch the graph.
244. The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.
246. symmetric with respect to the polar axis
248. symmetric with respect to the polar axis, symmetric with respect to the line
θ = π,
2 symmetric with respect to the pole
250. no symmetry
252. no symmetry
254. symmetric with respect to the pole 194
256. circle
258. cardioid
260. cardioid
262. one-loop/dimpled limaçon 195
264. one-loop/dimpled limaçon
266. inner loop/two-loop limaçon
268. inner loop/two-loop limaçon 196
270. inner loop/two-loop limaçon
272. lemniscate
274. lemniscate 197
276. rose curve
278. rose curve
280. Archimedes’ spiral
282. Archimedes’ spiral 198
284.
286.
288.
290. 199
292.
294. They are both spirals, but not quite the same.
296. Both graphs are curves with 2 loops. The equation with a coefficient of
θ has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to
4π to get a better picture.
298. When the width of the domain is increased, more petals of the flower are visible.
300. The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.
302. The graphs are spirals. The smaller the coefficient, the tighter the spiral.
304.
⎛ π ⎞ ⎛ 5π ⎞
⎝4, 3 ⎠, ⎝4, 3 ⎠
306.
⎛3 π ⎞ ⎛3 5π ⎞
⎝2 , 3 ⎠, ⎝2 , 3 ⎠
308.
⎛ π⎞
⎛ 3π ⎞
⎝0, 2 ⎠, (0, π), ⎝0, 2 ⎠, (0, 2π)
310.
⎛4
⎞ ⎛4
⎞
⎜ 8 , π ⎟, ⎜ 8 , 5π ⎟
⎝ 2 4⎠ ⎝ 2 4 ⎠
θ = 3π , 7π
4
4 since
r is squared
and at
312. a is the real part, b is the imaginary part, and
i = −1
314. Polar form converts the real and imaginary part of the complex number in polar form using
x = rcosθ and
y = rsinθ.
316.
z n = r n ⎛⎝cos(nθ) + isin(nθ)⎞⎠ It is used to simplify polar form when a number has been raised to a power. 200
318.
5 2
320.
38
322.
14.45
324.
4 5cis(333.4°)
326.
⎛ ⎞
2cis⎝π ⎠
6
328.
7 3 + i7
2
2
330.
−2 3 − 2i
332.
−1.5 − i 3 3
2
334.
4 3cis(198°)
336.
3 cis(180°)
4
338.
⎛
⎞
5 3cis⎝17π ⎠
24
340.
7cis(70°)
342.
5cis(80°)
344.
⎛ ⎞
5cis⎝π ⎠
3
346.
125cis(135°)
348.
9cis(240°)
350.
⎛
⎞
cis⎝3π ⎠
4
352.
3cis(80°), 3cis(200°), 3cis(320°)
354.
3
⎛ ⎞ 3
⎛ ⎞ 3
⎛
⎞
2 4cis⎝2π ⎠, 2 4cis⎝8π ⎠, 2 4cis⎝14π ⎠
9
9
9
356.
⎛ ⎞
⎛
⎞
2 2cis⎝7π ⎠, 2 2cis⎝15π ⎠
8
8
358. 201
360.
362.
364. 202
366.
Plot of 1-4i in the complex plane (1 along the real axis, -4 along the imaginary axis).
368.
3.61e −0.59i
370.
− 2 + 3.46i
372.
− 4.33 − 2.50i
373. A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For
example,
x = f (t) and
y = f (t).
375. Choose one equation to solve for
t, substitute into the other equation and simplify.
377. Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately
the equations are functions.
379.
y = − 2 + 2x
381.
y=3 x−1
2
383.
x = 2e
1−y
5
or
⎛ ⎞
y = 1 − 5ln⎝ x ⎠
2
385.
⎛y − 3 ⎞
x = 4log⎝
2 ⎠
387.
3
y
⎛y ⎞
x=⎝ ⎠ −
2
2
389.
y = x3
391.
⎛y ⎞
⎛x ⎞
⎝4 ⎠ + ⎝5 ⎠ = 1
2
2 203
393.
y2 = 1 − 1 x
2
395.
y = x 2 + 2x + 1
397.
⎞
⎛
3
y = ⎝x + 1 ⎠ − 2
2
399.
y = − 3x + 14
401.
y= x+3
403.
⎧x(t)
⎨
⎩y(t)
405.
⎧x(t)
⎨
⎩y(t)
407.
⎧x(t)
⎨
⎩y(t)
=t
= 2sint + 1
= t + 2t
=t
= 4cos t
;
= 6sin t Ellipse
409.
⎧x(t) = 10cost
⎨
;
⎩y(t) = 10sint
Circle
411.
⎧x(t)
⎨
⎩y(t)
413.
⎧x(t)
⎨
⎩y(t)
= − 1 + 4t
= − 2t
= 4 + 2t
= 1 − 3t
415. yes, at
t=2
417.
t
x
y
1
-3
1
2
0
7
3
5
17
419. answers may vary:
⎧x(t) = t − 1
⎧x(t) = t + 1
⎨
⎨
and
⎩y(t) = t 2
⎩y(t) = (t + 2) 2
421. answers may vary: ,
⎧x(t) = t
⎧x(t) = t + 2
⎨
and ⎨
2
⎩y(t) = t − 4t + 4
⎩y(t) = t 2
422. plotting points with the orientation arrow and a graphing calculator
424. The arrows show the orientation, the direction of motion according to increasing values of
t. 204
426. The parametric equations show the different vertical and horizontal motions over time.
428.
430.
432.
434.
436. 205
438.
440.
442.
444. 206
446.
448.
450.
452. 207
454.
456.
458. 208
460. There will be 100 back-and-forth motions.
462. Take the opposite of the
x(t) equation.
464. The parabola opens up.
466.
⎧x(t)
⎨
⎩y(t)
468.
470.
= 5cost
= 5sint 209
472.
474.
a = 4, b = 3, c = 6, d = 1
476.
a = 4, b = 2, c = 3, d = 3 210
478. 211
480.
482. The
y -intercept changes.
484.
⎛ ⎞
⎛ ⎞
y(x) = − 16⎝ x ⎠ + 20⎝ x ⎠
15
15
2
486.
⎧x(t) = 64tcos(52 ° )
⎨
⎩y(t) = − 16t 2 + 64tsin(52 ° )
488. approximately 3.2 seconds
490. 1.6 seconds
492. 212
494.
495. lowercase, bold letter, usually
u, v, w
497. They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a
magnitude of 1.
499. The first number always represents the coefficient of the
i, and the second represents the
j.
501.
〈7, − 5 〉
503. not equal
505. equal
507. equal
509.
7i − 3 j
511.
−6i − 2 j
513.
u + v = 〈−5, 5 〉, u − v = 〈−1, 3 〉, 2u − 3v = 〈0, 5 〉
515.
−10i – 4 j
517.
− 2 29 i + 5 29 j
29
29
519.
− 2 229 i + 15 229 j
229
229
521.
− 7 2i + 2 j
10
10 213
523.
|v| = 7.810, θ = 39.806 °
525.
|v| = 7.211, θ = 236.310°
527.
−6
529.
−12
531. 214 215
533.
535.
537.
539.
541.
〈4, 1 〉
543.
v = − 7i + 3 j 216
545.
3 2i + 3 2 j
547.
i − 3j
549. a. 58.7; b. 12.5
551.
x = 7.13 pounds,
y = 3.63 pounds
553.
x = 2.87 pounds,
y = 4.10 pounds
555. 4.635 miles, 17.764° N of E
557. 17 miles. 10.318 miles
559. Distance: 2.868. Direction: 86.474° North of West, or 3.526° West of North
561. 4.924°. 659 km/hr
563. 4.424°
565.
(0.081, 8.602)
567. 21.801°, relative to the car’s forward direction
569. parallel: 16.28, perpendicular: 47.28 pounds
571. 19.35 pounds, 231.54° from the horizontal
573. 5.1583 pounds, 75.8° from the horizontal
Review Exercises
574. Not possible
576.
C = 120°, a = 23.1, c = 34.1
578. distance of the plane from point
A : 2.2 km, elevation of the plane: 1.6 km
580.
B = 71.0°, C = 55.0°, a = 12.8
582. 40.6 km 217
584.
586.
(0, 2)
588.
(9.8489, 203.96°)
590.
r=8
592.
x 2 + y 2 = 7x
594.
y= −x
596. symmetric with respect to the line
θ=π
2 218
598.
600.
602. 5
604.
⎛
⎞
cis⎝− π ⎠
3
606.
2.3 + 1.9i
608.
⎛ ⎞
60cis⎝π ⎠
2
610.
⎛
⎞
3cis⎝4π ⎠
3
612.
⎛
⎞
25cis⎝3π ⎠
2
614.
⎛ ⎞
⎛ ⎞
5cis⎝3π ⎠, 5cis⎝7π ⎠
4
4
616. 219
618.
x2 + 1 y = 1
2
620.
⎧x(t)
⎨
⎩y(t)
= − 2 + 6t
= 3 + 4t
622.
y = − 2x 5
624.
a.
⎧x(t) = (80cos(40°))t
⎨
⎩y(t) = − 16t 2 + ⎛⎝80sin(40°)⎞⎠t + 4
b. The ball is 14 feet high and 184 feet from where it was launched.
c. 3.3 seconds
626. not equal
628. 4i
630.
− 3 10
10 i
− 10
10 j
632. Magnitude:
3 2, Direction:
225°
634.
16 220
636.
Practice Test
638.
α = 67.1°, γ = 44.9°, a = 20.9
640.
1712 miles
642.
⎛
⎝
1, 3⎞⎠
644.
y= −3 221
646.
648.
106
650.
−5 + i 5 3
2
2
652.
4cis(21°)
654.
2 2cis(18°), 2 2cis(198°)
656.
y = 2(x − 1) 2
658.
660. −4i − 15j
662.
2 13 i + 3 13 j
13
13
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