SOLUTION: Use trigonometric identities to find the exact value of tan theta if sin theta=1/3 and tan theta<0

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Question 903166: Use trigonometric identities to find the exact value of tan theta if sin theta=1/3 and tan theta<0
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Since tan%28theta%29%3C0 then theta is either in QII or QIV.
Since sin%28theta%29 is given as a positive number 1%2F3, then
theta is in QII



So

(1)   tan%28theta%29=1%2F%283cos%28theta%29%29

But we will need cos%28theta%29, which is negative in QII.

sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1
%281%2F3%29%5E2%2Bcos%5E2%28theta%29=1
1%2F9%2Bcos%5E2%28theta%29=1
1%2B9cos%5E2%28theta%29=9
9cos%5E2%28theta%29=8
cos%5E2%28theta%29=8%2F9
cos%28theta%29=-sqrt%288%2F9%29  <-- we take the negative square root since cosine is negative in QII
cos%28theta%29=-sqrt%288%29%2F3
cos%28theta%29=-sqrt%284%2A2%29%2F3
cos%28theta%29=-2sqrt%282%29%2F3

Now go back to equation (1)



Edwin