How do you determine #sintheta# given #sectheta=-7/5, 180^circ<theta<270^circ#?

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Answer 1 · 2016-11-16T01:28:33

Please see the explanation.

The secant is the reciprocal of the cosine:

#cos(theta) = 1/sec(theta) = -5/7#

This is a variant of #sin^2(theta) + cos^2(theta) = 1#

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

Substitute #25/49# for #cos^2(theta)#

#sin(theta) = +-sqrt(1 - 25/49)#

#sin(theta) = +-sqrt(24/49)#

#sin(theta) = +-sqrt(24)/7#

We are told that we are in the third quadrant so use the negative value:

#sin(theta) = -sqrt(24)/7#