Question

How do I evaluate sin from cos and use symmetry arguments?

If `cos(theta)=0.8, and 270°< theta<360°`
a) `evaluate sin(theta)` and show it on a unit circle
b) Using symmetry arguments evaluate `cos(theta-180°)`
c) Confirm the result using the trigonometric identities

Answer 1

sin t = - 0.6
cos (x - 180) = - cos x

Explanation:

cos t = 0.8 , and t lies in Quadrant 4.
a. Find sin t by using trig identity: `sin^2 t + cos^2 t = 1`
In this case:
`sin^2 t = 1 - 0.64 = 0.36`
`sin t = +- 0.6`
Since t lies in Quadrant 4, so, sin t is negative
sin t = -0.6
Calculator and unit circle give 2 solutions for t
`t = - 36^@87`, and `t = 180 - (-36.87) = 216^@87`
b. Compare the arc x and the arc (x - 180). They are symmetrical
over the origin O. The segment cos x and the segment cos (x - 180) are symmetrical over the origin O. Therefor,
cos (x - 180) = - cos x.
c. Use trig identity: cos (a - b) = cos a.cos b + sin a.sin b
In this case:
cos a = cos x --> cos b = cos 180 = -1 --> sin b = sin 180 = 0.
Therefor,
cos (x - 180) = - cos x