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1 Answer
Apr 28, 2018

tantheta=(7sqrt15)/(64) sectheta=(-8sqrt15)/15

Explanation:

We're told that sintheta is negative; however, cottheta=costheta/sintheta is positive.

This tells us that both sine and cosine must be negative (the negatives cancel out to give a positive cotangent), so we're working in the fourth quadrant.

Then, tantheta=sintheta/costheta will also be positive.

Furthermore, since sectheta=1/costheta, sectheta will also be negative as a result of the negative cosine.

Keeping this in mind, we'll continue.

Recall the identity

sin^2theta+cos^2theta=1

sintheta=(-7/8), sin^2theta=(-7/8)^2=49/64

Thus,

49/64+cos^2theta=64/64

cos^2theta=(64-49)/64

cos^2theta=15/64

costheta=+-sqrt(15/64)=+-sqrt15/8

We want the negative result:

costheta=-sqrt15/8

Then,

sectheta=1/costheta=1/(sqrt15/8)=-8/sqrt15

Rationalizing the denominator (multiply by sqrt15/sqrt15), we get

sectheta=-(8sqrt15)/15

We can now find the tangent:

tantheta=sintheta/costheta

tantheta=(cancel-7/8)/(cancel-(8sqrt15)/15)

tantheta=7/8*15/(8sqrt15)

tantheta=105/(64sqrt15)

Rationalizing, we get

tantheta=(105sqrt15)/(960)

tantheta=(7sqrt15)/(64)