By using trigonometric identities find the value of sin A if tan A equals a half?

Answer 1

If tan A = 1/2, then sin A = ?

We use the Pythagorean identity 1 + cot2 A = csc2 A to find csc A, and then the reciprocal identity sin A = 1/csc A to find sin A.

tan A = 1/2 (since tan A is positive, A is in the first or the third quadrant)

cot A = 1/tan A = 1/(1/2) = 2

1 + cot2 A = csc2 A

1 + (2)2 = csc2 A

5 = csc2 A

√5 = csc A (when A is in the first quadrant)

1/√5 = sin A

√5/5 = sin A

If A is in the third quadrant, then sin A = -√5/5.