Question

If sin(x) = 5/13, and x is in quadrant 1, then tan(x/2) equals what?

Answer 1

1/5

I just took the test and got it correct!

Answer 2

`x` is in quadrant I, so
`0<x<\frac\pi2`, which means
`0<\frac x2<\frac\pi4`, so
`\frac x2` belongs to the same quadrant.

Now,


`\tan^2\frac x2=(\sin^2\frac x2)/(\cos^2\frac x2)=\frac{\frac{1-\cos x}2}{\frac{1+\cos x}2}=(1-\cos x)/(1+\cos x)`

Since
`\sin x=\frac5{13}`, it follows that


`\cos^2x=1-\sin^2x\implies \cos x=\pm\sqrt{1-\left(\frac5{13}\right)^2}=\pm(12)/(13)`

Since
`x` belongs to the first quadrant, you take the positive root (
`\cos x>0` for
`x` in quadrant I). Then


`\tan\frac x2=\pm\sqrt{(1-(12)/(13))/(1+(12)/(13))}`


`\tan x` is also positive for
`x` in quadrant I, so you take the positive root again. You're left with


`\tan\frac x2=\frac15`