A tree 100 ft tall casts a shadow 120 ft long. Find the angle of elevation of the sun. in .NET

Draw QR Code in .NET A tree 100 ft tall casts a shadow 120 ft long. Find the angle of elevation of the sun.

3.11 A tree 100 ft tall casts a shadow 120 ft long. Find the angle of elevation of the sun.
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In Fig. 3.15, CB 100, AC 120, and we want to find A.
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Fig. 3.15
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Manual Solution tan A tan A tan A A CB AC 100 120 0.83 40
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Calculator Solution tan A tan A tan A A A CB AC 100 120 0.833333 39.8056 40
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(Since tan 40 has the closest value to 0.83, we used A
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40 .)
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3.12 A ladder leans against the side of a building with its foot 12 ft from the building. How far from the ground is the top of the ladder and how long is the ladder if it makes an angle of 70 with the ground
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From Fig. 3.16, tan A CB/AC; then CB AC tan A 12 tan 70 12(2.75) 33. The top of the ladder is 33 ft above the ground. Manual: sec A AB/AC; then AB AC sec A 12 sec 70 12(2.92) 35.04. The calculator solution procedure is the same.
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CHAPTER 3 Trigonometric Functions of an Acute Angle
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Fig 3.16
Calculator: cos A
AC/AB; then AB
AC/(cos A)
12/(cos 70 )
12/0.342020
The ladder is 35 ft long.
3.13 From the top of a lighthouse 120 m above the sea, the angle of depression of a boat is 15 . How far is the boat from the lighthouse
In Fig. 3.17, the right triangle ABC has A Manual: cot A Calculator: tan A AC/CB and AC CB/AC and AC CB cot A 15 and CB 120 cot 15 120. 120(3.73) 447.6. 447.846.
CB/(tan A)
120/(tan 15 )
120/0.267949
The boat is 448 m from the lighthouse.
Fig. 3.17
3.14 Find the length of the chord of a circle of radius 20 cm subtended by a central angle of 150 .
In Fig. 3.18, OC bisects / AOB. Then BC Manual: In OAC, sin / COA BA 2(19.4) 38.8. Calculator: AC OA sin / COA AC and OCA is a right triangle. OA sin / COA 20 sin 75 20(0.97) 38.6370. 19.4; AC/OA and AC 20 sin 75
20(0.965926)
19.3185; BA
2(19.3185)
The length of the chord is 39 cm.
B C 75 O 20 A
Fig. 3.18
CHAPTER 3 Trigonometric Functions of an Acute Angle
3.15 Find the height of a tree if the angle of elevation of its top changes from 20 to 40 as the observer advances 75 ft toward its base. See Fig. 3.19.
Fig. 3.19
In the right triangle ABC, cot A In the right triangle DBC, cot D Manual:
AC/CB; then AC DC/CB; then DC DC CB(cot 20
CB cot A or DC CB cot 40 . 75 1.19) CB
CB cot 20 .
CB cot 20 CB(2.75
CB cot 40 75 75 75/1.56 48.08 2.74748 1.19175
cot 40 )
and Calculator: cot 20 cot 40 CB(cot 20 CB(2.74748 CB The tree is 48 ft tall. 1/tan 20 1/tan 40
1/0.363970 1/0.839100 75 75 75 48.2089
cot 40 ) 1.19175) 75/1.55573
CB(1.55573)
3.16 A tower standing on level ground is due north of point A and due west of point B, a distance c ft from A. If the angles of elevation of the top of the tower as measured from A and B are and , respectively, find the height h of the tower.
AC In the right triangle ACD of Fig. 3.20, cot h cot and BC h cot . Since ABC is a right triangle, (AC)2 (BC)2 h AC/h; and in the right triangle BCD, cot c2 h2(cot )2 c 2(cot a)2 (cot b)2 h2(cot )2 and BC/h. Then
Fig. 3.20
3.17 If holes are to be spaced regularly on a circle, show that the distance d between the centers of two successive holes is given by d 2r sin (180 /n), where r the radius of the circle and n the number of holes. Find d when r 20 in and n 4.
CHAPTER 3 Trigonometric Functions of an Acute Angle
In Fig. 3.21, let A and B be the centers of two consecutive holes on the circle of radius r and center O. Let the bisector of the angle O of the triangle AOB meet AB at C. In right triangle AOC, sin /AOC AC r
1 2d
d 2r
Fig. 3.21
Then
2r sin / AOC
1 360 2r sin 2 a n b
2r sin 1 / AOB 2 180 2r sin n 20 22 in.
When r
20 and n
4, d
2 20 sin 45
2 20 ( 22>2)
SUPPLEMENTARY PROBLEMS
3.18 Find the exact values of the trigonometric functions of the acute angles of the right triangle ABC, given: (a) a 3, b 1; (b) a 2, c 5; (c) b 27, c 4
Ans. Answers are in the order sine, cosine, tangent, cotangent, secant, and cosecant. (a) A: 3> 210 B: 1> 210 3 210>10, 1> 210 210>10, 3> 210 210>10, 3, 1>3 , 210, 210>3; 3 210>10, 1>3 , 3, 210>3, 210 5 221>21, 5>2; 5 221>21 2 221>21, 5>2 , 5> 221 3 27>7, 4>3, 4 > 27
(b) A: 2>5, 221>5, 2> 221 (c) A: 3>4, 27>4, 3> 27
2 221>21, 221>2, 5> 221 3 27>7, 27>3, 4> 27
B: 221>5, 2>5, 221>2, 2> 221 B: 27>4, 3>4, 27>3, 3> 27 3.19 Which is the greater (a) sin 55 or cos 55 (b) sin 40 or cos 40 Ans. (a) sin 55 , 3.20 (b) cos 40 ,
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