How do you find the exact value of cos(theta) if sin(theta)=-2/3?

2 Answers
Lucy
Apr 2, 2018

cos #theta#= #(sqrt5)/3#
or it could be cos #theta#= #(sqrt5)/-3#

Explanation:

Since sin #theta# is negative, it can be in the third or fourth quadrant

Drawing your right-angled triangle, place your #theta# in one of three corners. Your longest side will be 3 and the side opposite the #theta# will be -2. Finally, using Pythagoras theorem, your last side should be #sqrt5#

Now, if your triangle was in the third quadrant, you would have
cos #theta#= #(sqrt5)/-3# since cosine is negative in the third quadrant

But if your triangle was in the fourth quadrant, you would have
cos #theta#= #(sqrt5)/3# since cosine is positive in the fourth quadrant

sankarankalyanam
Apr 2, 2018

#color(indigo)(cos theta = +- sqrt5 / 3#

Explanation:

#sin theta = -2/3#

https://hononegah.learning.powerschool.com/hhearn/2014-2015honorspre-calculus/cms_page/view/16304893

#sin^2 theta = (-2/3)^2 = 4/9#

But #cos^2 theta = 1 - sin^2 theta = 1 - 4/9 = 5/9#

#:. cos theta = sqrt(5/9) = +- sqrt5 / 3#

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