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3.1 Find the trigonometric functions of the acute angles of the right triangle ABC, Fig. 3.6, given b and c 25.
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b2 sin A cos A tan A
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7. Then 7 25 cot A sec A csc A adjacent side opposite side hypotenuse adjacent side hypotenuse opposite side 24 7 25 24 25 7
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opposite side hypotenuse adjacent side hypotenuse opposite side adjacent side
24 25 7 24
and sin B cos B tan B 24/25 7/25 24/7 cot B sec B csc B 7/24 25/7 25/24
CHAPTER 3 Trigonometric Functions of an Acute Angle
3.2 Find the values of the trigonometric functions of the acute angles of the right triangle ABC, Fig. 3.7, given a 2 and c 2 25.
Fig. 3.7
Since b2
c2 sin A cos A tan A
(2 25)2
(2)2
20 cos B
16, b
4. Then cot A sec A csc A 4/2 2 tan B 25>2 25 csc B sec B
2/2 25 4/2 25 2/4 1/2
25>5 2 25>5 cot B
sin B
2 25>4 2 25>2
3.3 Find the values of the trigonometric functions of the acute angle A, given sin A
Construct the right triangle ABC, Fig. 3.8, with a sin A cos A tan A 3/7 2 210>7 3/2 210 3 210>20 3, c 7, and b cot A sec A csc A 27
3/7.
2 210 units. Then
2 210>3 7/2 210 7/3 7 210>20
Fig. 3.8
3.4 Find the values of the trigonometric functions of the acute angle B, given tan B
Refer to Fig. 3.9. Construct the right triangle ABC having b 15 and a therefore a right triangle with b 3 and a 2 will serve equally well.) Then c 2a2 sin B cos B tan B b2 2102 152 3 213>13 2 213>13 5 213 and cot B sec B csc B 2/3 5 213>10 5 213>15 213>2 213>3 15/5 213 10/5 213 15/10 3/2
3 2,
10 units. (Note that 1.5
213>3
Fig. 3.9
CHAPTER 3 Trigonometric Functions of an Acute Angle
3.5 If A is acute and sin A
2x/3, determine the values of the remaining functions.
2x < 3 and c 3, as in Fig. 3.10
Construct the right triangle ABC having a
Fig. 3.10
Then b
2c2 sin A
a2 2x 3 29
29 cos A 4x2 2x
4x2 and 29 3 sec A 4x2 tan A 3 29 4x2 2x 29 4x2 2x 29 4x2 9 4x2 3 2x
cot A
3 29 4x2 csc A 9 4x2
3.6 If A is acute and tan A
x/1, determine the values of the remaining functions.
x and b cos A csc A 1, as in Fig. 3.11. Then c 1 2x2 2x2 x 1 1 2x2 1 x2 1
Construct the right triangle ABC having a sin A cot A x 2x2 1 x sec A 1 x 2x2 1 x2 1 2x2 1
tan A x
1 and
Fig. 3.11
3.7 If A is an acute angle: (a) Why is sin A < 1 (b) When is sin A cos A (c) Why is sin A < csc A
In the right triangle ABC: (a) Side a < side c; therefore sin A (b) Sin A (d) Sin A (f) Tan A cos A when a/c a/c, tan A (c) Sin A < 1 (above) and csc A a/c < 1. b, A B, and A 45 . 1/sin A > 1. A and A < 45 .
(d) Why is sin A < tan A (e) When is sin A < cos A (f) When is tan A > 1
b/c; then a
a/b, and b < c; therefore a/c < a/b or sin A < tan A.
(e) Sin A < cos A when a < b; then A < B or A < 90
a/b > 1 when a > b; then A > B and A > 45 .
CHAPTER 3 Trigonometric Functions of an Acute Angle
3.8 Find the exact values of the trigonometric functions of 45 . (See Fig. 3.12.)
Fig. 3.12
In any isosceles right triangle ABC, A and sin 45 cos 45 tan 45 1> 22 1> 22 1/1 1
45 and a
b. Let a cot 45 sec 45 csc 45 1
1; then c
1 2 22 1 2 22
22 22
3.9 Find the exact values of the trigonometric functions of 30 and 60 . (See Fig. 3.13.)
Fig. 3.13
In any equilateral triangle ABD, each angle is 60 . The bisector of any angle, like B, is the perpendicular bisector of the opposite side. Let the sides of the equilateral triangle be of length 2 units. Then in the right triangle 222 12 23. ABC, AB 2, AC 1, and BC sin 30 cos 30 tan 30 1/2 cos 60 sin 60 23>3 cot 60 cot 30 sec 30 csc 30 23 2> 23 2 tan 60 2 23>3 csc 60 sec 60 23>2 1> 23
(NOTE: In Probs. 3.10 to 3.15 two solution procedures are shown, one for manual solution and one for calculator solution, whenever the two are different. Which one you use depends upon your access to a calculator during your problem-solving work. If your access to a calculator is restricted, then focus only on the manual solutions. In the calculator solutions, steps are shown to illustrate the procedures rather than as a guide to work steps that need to be shown. The steps shown in each solution are to allow you to see all the details of the procedure used.)
3.10 When the sun is 20 above the horizon, how long is the shadow cast by a building 50 m high
Fig. 3.14
CHAPTER 3 Trigonometric Functions of an Acute Angle
In Fig. 3.14, A 20 , CB 50, and AC is to be found. Manual Solution cot A AC AC AC AC AC AC CB CB cotA 50 cot 20 50(2.75) 137.5 138 m Calculator Solution tan A AC AC AC AC AC CB AC CB tanA 50 tan20 50 0.363970 137.374 137 m
(NOTE: The difference in the answers for the two procedures is because cot 20 was rounded to two decimal places in the table. Each answer is the correct one for that procedure.)
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