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

If cos(theta)=0.8, and 270°< theta<360°  a) $e v a l u a t e \sin \left(\theta\right)$ and show it on a unit circle b) Using symmetry arguments evaluate cos(theta-180°) c) Confirm the result using the trigonometric identities

Jun 27, 2018

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:
${\sin}^{2} t = 1 - 0.64 = 0.36$
$\sin t = \pm 0.6$
Since t lies in Quadrant 4, so, sin t is negative
sin t = -0.6
Calculator and unit circle give 2 solutions for t
$t = - {36}^{\circ} 87$, and $t = 180 - \left(- 36.87\right) = {216}^{\circ} 87$
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