SOLUTION: If sin (theta)=-9/13 and cos (theta)>0, then find the exact value (no decimals) of cos (theta).

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Question 113994: If sin (theta)=-9/13 and cos (theta)>0, then find the exact value (no decimals) of cos (theta).
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember sine is the ratio of the opposite side over the hypotenuse (ie sin%28theta%29=opposite%2Fhypotenuse). So think of 9 as the length of the opposite side and 13 as the length of the hypotenuse. Also, since the ratio is negative, the opposite side is negative (the hypotenuse is never negative). So we basically have this triangle set up:




Since we can see that the triangle has legs of 9 and x with a hypotenuse of 13, we can use Pythagoreans theorem to find the unknown side.


Pythagoreans theorem:

a%5E2%2Bb%5E2=c%5E2 where a and b are the legs of the triangle and c is the hypotenuse



9%5E2%2Bx%5E2=13%5E2 Plug in a=9, b=x, and c=13. Now lets solve for x


8+1+%2B++x++%5E+2+=+1+6+9 Square each individual term



+x++%5E+2+=+1+6+9+-+8+1 Subtract 81 from both sides


+x++%5E+2+=+8+8 Combine like terms


s+q+r+t+%28++x++%5E+2+%29+=+s+q+r+t+%28+8+8+%29 Take the square root of both sides


x=2%2Asqrt%2822%29 Simplify the square root



Now let's find cos(theta)


cosine is defined as the adjacent (which is x) over the hypotenuse (which is 13)

So cos%28theta%29=adjacent%2Fhypotenuse=x%2F13=2%2Asqrt%2822%29%2F13