Question

if 8 sins A = 4 + cos A then find the value of tan A

Answer 1

Step-by-step explanation:

Solution :

SinA : CosA =4 :8 = 1 : 2

tanA = SinA : CosA = 1: 2.

Hence tanA = 1 : 2.

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Answer 2

if 8 sins A = 4 + cos A then the value of tan A = - 5/12, 3/4

Given,

8 sin A = 4 + cos A

upon dividing the both sides by cos A, we get,

8 sin A/cos A = 4/cos A + cos A/cos A

we know the trigonometric relations, sin/cos = tan and 1/cos = sec

we have,

8 tan A = 4 sec A + 1

8 tan A - 1 = 4 sec A

squaring on both sides, we get,

(8 tan A - 1)² = (4 sec A)²

8² tan² A + 1² - 2 × 8 tan A × 1 = 4² sec² A

64 tan² A + 1 - 16 tan A = 16 sec² A

as we know that sec² A = 1 + tan² A, we get,

64 tan² A + 1 - 16 tan A = 16 ( 1 + tan² A )

64 tan² A + 1 - 16 tan A = 16 + 16 tan² A

64 tan² A - 16 tan² A  - 16 tan A = 16 - 1

48 tan² A  - 16 tan A = 15

48 tan² A  - 16 tan A - 15 = 0

solving the above quadratic equation, we get,

(12 tan A + 5) (4 tan A - 3) = 0

⇒ tan A = - 5/12  and tan A = 3/4

∴  tan A = - 5/12, 3/4