If sin theta = 1/4, theta in quadrant 2, how do you find the exact value of sin(theta - pi/3)?

1 Answer
Jun 18, 2018

sin (t - pi/3) = (1 + 3sqrt5)/8

Explanation:

sin t = 1/4. Find cos t
cos^2 t = 1 - sin^2 t = 1 - 1/16 = 15/16
cos t = - sqrt15/4 (because t lies in Quadrant 2)
From trig identity, we get:
sin (t - pi/3) = sin t.cos (pi/3) - sin (pi/3).cos t (1)
Note.
sin t = 1/4 --> cos t = - sqrt15/4 -->
cos (pi/3) = 1/2 --> sin (pi/3) = sqrt3/2
Equation (1) becomes:
sin (t - pi/3) = 1/8 + (sqrt3.sqrt15)/8 = (1 + 3sqrt5)/8
Check by calculator.
sin t = 0.25 --> t = 162^@52 --> (t - 60) = 105^@22
sin (t - 60) = 0.96
(1 + 3sqrt5)/8 = 0.96. Proved.