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CHAPTER 8 Trigonometry
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x (a) If x is the height of the part of the building above the transit [Fig. 8-8(a)], then tan 21 and 200 x 200 tan 21 200(0.3839) 77 ft. Thus, the height of the building is h x 5 77 5 82 ft. h (b) If h is the height of the tree [Fig. 8-8(b)], then we have tan 42 and h 25 tan 42 25(0.9004) 25 23 ft.
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Fig. 8-8
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Using both an angle of elevation and an angle of depression Standing at the top of a lighthouse 200 ft high, a lighthouse keeper sighted both an airplane and a ship directly beneath the plane. The angle of elevation of the plane measured 25 ; the angle of depression of the ship measured 32 . Find (a) the distance d of the boat from the foot of the lighthouse, to the nearest 10 ft; (b) the height of the plane above the water, to the nearest 10 ft.
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(a) See Fig. 8-9. In nIII, tan 58 (b) In nI, tan 25 d and d 200 200 tan 58 200(1.6003) 320 ft.
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x 320 and x 320(0.4663) the airplane is 200 150 350 ft.
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150 ft. Since the height of the tower is 200 ft, the height of
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Fig. 8-9
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Using two angles of depression An observer on the top of a hill 250 ft above the level of a lake sighted two boats directly in line. Find, to the nearest foot, the distance between the boats if the angles of depression noted by the observer measured 11 and 16 .
CHAPTER 8 Trigonometry
Solutions
In nAB C of Fig. 8-10, m B AC In nABC, m BAC Hence, BB 90 16 90 11 74 . Then CB 79. Then CB 250 tan 74 . 250(5.1446 3.4874) 414 ft. 250 tan 79 .
CB CB
250(tan 79 tan 74 )
Fig. 8-10
SUPPLEMENTARY PROBLEMS
8.1. Using the table of trigonometric functions at the end of the book, find (a) sin 25 , sin 48 , sin 59 , and sin 89 (b) cos 15 , cos 52 , cos 74 , and cos 88 (c) tan 4 , tan 34 , tan 55 , and tan 87 (d) which trigonometric ratios increase as the measure of the angle increases from 0 to 90 (e) which trigonometric ratio decreases as the measure of the angle increases from 0 to 90 (f) which trigonometric ratio has values greater than 1 8.2. Using the table at the end of the book, find the angle for which (a) sin x (b) sin A 8.3. 0.3420 0.4848 (c) sin B (d) cos A 0.9455 0.9336 (e) cos y (f) cos Q 0.7071 0.3584 (g) tan W (h) tan B 0.3443 2.3559 (8.2) 3 8 23 2 2 7 23 10 (8.3) (8.1) (8.1)
Using the table at the end of the book, find the measure of x to the nearest degree if (a) sin x (b) sin x (c) sin x (d) cos x 0.4400 0.7280 0.9365 0.9900 (e) cos x (f) cos x (g) tan x (h) tan x 0.7650 0.2675 0.1245 0.5200 (i) tan x (j) sin x (k) sin x (l) cos x 5.5745 11 50 22 2 13 25 (m) cos x (n) cos x (o) tan x (p) tan x
In each right triangle of Fig. 8-11, find sin A, cos A, and tan A.
Fig. 8-11
CHAPTER 8 Trigonometry
Find m A to the nearest degree in each part of Fig. 8-12.
(8.4)
Fig. 8-12
Find m B to the nearest degree if (a) b (d) a 3 and b 23 in Fig. 8-13.
67 and c
100; (b) a
14 and c
50; (c) a
22 and b
55; (8.4)
Fig. 8-13
Fig. 8-14
Making use of a square with a side of 1 (Fig. 8-14), show that (a) the diagonal c (c) sin 45 cos 45 0.707.
22; (b) tan 45
1; (8.5)
To the nearest degree, find the measure of each acute angle of any right triangle whose sides are in the ratio of (a) 5:12:13; (b) 8:15:17; (c) 7:24:25; (d) 11:60:61. (8.4) In each triangle of Fig. 8-15, solve for x and y to the nearest integer. (8.5)
Fig. 8-15
A ladder leans against the side of a building and makes an angle measuring 70 with the ground. The foot of the ladder is 30 ft from the building. Find, to the nearest foot, (a) how high up on the building the ladder reaches; (b) the length of the ladder. (8.6) To find the distance across a swamp, a surveyor took measurements as shown in Fig. 8-16. AC is at right angles to BC. If m A 24 and AC 350 ft, find the distance BC across the swamp. (8.6)
Fig. 8-16
Fig. 8-17