How do you find #cos(theta)# if #sin(theta)= 3/5# and #90 < theta < 180#?

1 Answer
VNVDVI
Mar 31, 2018

#costheta=-4/5#

Explanation:

#90<theta<180# implies that we are in the second quadrant, where the cosine is negative and the sine is positive. So, keep in mind, #costheta# will be negative.

Recall the identity

#sin^2theta+cos^2theta=1#

This tells us that

#cos^2theta=1-sin^2theta -> costheta=+-sqrt(1-sin^2theta)#

In our situation, as we are in the second quadrant, #costheta=-sqrt(1-sin^2theta)#

We have #sintheta=3/5, sin^2theta=(3/5)^2=9/25#

So,

#costheta=-sqrt(1-9/25)#
#costheta=-sqrt(25/25-9/25)#

#costheta=-sqrt(16/25)=-4/5#

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