172
25.3
TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE
[CHAP. 25
Find the values of the trigonometric functions of the acute angle A, given sin A ¼ 37. pffiffiffiffiffiffiffiffiffi pffiffiffi Construct the right triangle ABC having a ¼ 3, c ¼ 7, and b ¼ 72 32 ¼ 2 10 units. See Fig. 25-5. Then sin A ¼ 37
pffiffiffi 2 10 7 pffiffiffi 3 3 10 tan A ¼ pffiffiffi ¼ 20 2 10
cos A ¼
pffiffiffi 2 10 3 pffiffiffi 7 7 10 sec A ¼ pffiffiffi ¼ 20 2 10
cot A ¼
csc A ¼ 73
Fig. 25-5 25.4
Find the values of the trigonometric functions of the acute angle B, given tan B ¼ 1:5. Refer to Fig. 25-6. Construct the right triangle ABC having b ¼ 15 and a ¼ 10 units. (Note that 1:5 ¼ 32 and a right triangle with b ¼ 3, a ¼ 2 will serve equally well.)
Fig. 25-6 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Then c ¼ a2 þ b2 ¼ 102 þ 152 ¼ 5 13 and pffiffiffi 15 3 13 sin B ¼ pffiffiffi ¼ 13 5 13 pffiffiffi 10 2 13 cos B ¼ pffiffiffi ¼ 13 5 13 3 tan B ¼ 15 10 ¼ 2
25.5
cot B ¼ 23 pffiffiffi pffiffiffi 13 5 13 ¼ 2 10 pffiffiffi pffiffiffi 13 5 13 csc B ¼ ¼ 3 15 sec B ¼
If A is acute and sin A ¼ 2x=3, determine the values of the remaining functions. Construct the right triangle ABC having a ¼ 2x < 3 and c ¼ 3, as in Fig. 25-7. pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Then b ¼ c2 a2 ¼ 9 4x2 and pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2x 2x 9 4x2 9 4x2 cos A ¼ ; tan A ¼ pffiffiffiffiffiffiffiffiffiffi ; ; cot A ¼ sin A ¼ ; 2 3 3 2x 9 4x 3 sec A ¼ pffiffiffiffiffiffiffiffiffiffi ; 9 4x2
csc A ¼
3 : 2x