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

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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

If

a)

b) Using symmetry arguments evaluate

c) Confirm the result using the trigonometric identities

##### 1 Answer

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:

In this case:

Since t lies in Quadrant 4, so, sin t is negative

sin t = -0.6

Calculator and unit circle give 2 solutions for t

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