Same angle trigonometry function example
Question:
Given: sin a = 1/3, in which the angle a is in the range of (pi/2, pi). Find the value of cos (a - pi/6).
Solution:
The initial side of the angle alfa lies in the positive x-axis. The terminal side of the angle alfa lies in between pi/2 to pi, which is in Quadrant2. In Quadrant2, sin a > 0 and cos a < 0.
- cos (a - pi/6)
- = cos a cos pi/6 + sin a sin pi/6
We use formula: cos (a + b) = cos a cos b - sin a sin b. Note: cos pi/6 = cos 30o = sin 60o = (square root of 3)/2. sin pi/6 = sin 30o = 1/2
- cos (a - pi/6)
- = [(square root of 3)/2] cos a + (1/2) sin a
Now we are going to find cos a
- sin2a + cos2a = 1
- cos2a = 1 - sin2a
Because in Quadrant2, cos a < 0
- cos a = - square root of (1 - sin2a)
- given: sin a = 1/3
- cos a = - square root of [1 - (1/3)2]
- = - square root of (1 - 1/9)
- = - square root of 8/9
- = - 2 (square root of 2)/3
- cos (a - pi/6)
- = (square root of 3)/2 × (- 2 square root of 2/3) + (1/2)(1/3)
- = 1/6 - square root of 6/3
- = 1/6 - 2/2 × (square root of 6)/3
- = 1/6 - 2 (square root of 6)/6
- = (1/6)(1 - 2 square root of 6)
- = - 0.65
Therefore, the value of cos (a - pi/6) is - 0.65. Watch the video for more details.