SOLUTION: find the value for sin(x/2) if the following conditions hold tan(x)=-4/3 and x is in quadrant IV

Question 1035159: find the value for sin(x/2) if the following conditions hold tan(x)=-4/3 and x is in quadrant IV
Found 2 solutions by solver91311, Alan3354:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Half-angle formula:

But that requires that we know and we only have and that

However, we also know that

So

Then, using the Pythagorean Identity:

And then

But since

Just plug this value into the half-angle formula and do the arithmetic. Don't forget to rationalize your denominator. Also, half of a QIV angle is in QII, so the sine is positive.

John

My calculator said it, I believe it, that settles it

Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
find the value for sin(x/2) if the following conditions hold tan(x)=-4/3 and x is in quadrant IV
---------------
tan = y/x
r = sqrt(y^2 + x^2) = 5
sin(x) = y/r = -4/5
------------------
sin^2(x/2) = (1 - cos(x))/2 ***** Much simpler than the sin(x/2)
cos(x) = x/r = 3/5 --- cosine is + in Q4
---
sin^2(x/2) = (2/5)/2 = 1/5
sin(x/2) = sqrt(1/5) = sqrt(5)/5
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