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galina1969 [7]
3 years ago
12

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

Mathematics
1 answer:
Rufina [12.5K]3 years ago
4 0
x is in quadrant I, so 0, which means 0, so \dfrac x2 belongs to the same quadrant.

Now,

\tan^2\dfrac x2=\dfrac{\sin^2\frac x2}{\cos^2\frac x2}=\dfrac{\frac{1-\cos x}2}{\frac{1+\cos x}2}=\dfrac{1-\cos x}{1+\cos x}

Since \sin x=\dfrac5{13}, it follows that

\cos^2x=1-\sin^2x\implies \cos x=\pm\sqrt{1-\left(\dfrac5{13}\right)^2}=\pm\dfrac{12}{13}

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

\tan\dfrac x2=\pm\sqrt{\dfrac{1-\frac{12}{13}}{1+\frac{12}{13}}}

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

\tan\dfrac x2=\dfrac15
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