Question

Find the exact values of sin(u/2) cos(u/2) and tan(u/2) given sin u=24/25 and pie/2 < u < pie

Answer 1

Recall the double/half angle formulas:

$\cos^2\dfrac x2=\dfrac{1+\cos x}2$

$\sin^2\dfrac x2=\dfrac{1-\cos x}2$

We're given $\sin u=\frac{24}{25}$, and since $u$ is between π/2 and π, we expect $\cos u$ to be negative. So from the Pythagorean identity, we find

$\sin^2u+\cos^2u=1\implies\cos u=-\sqrt{1-\sin^2u}=-\dfrac7{25}$

Also, we know $\frac u2$ will fall between π/4 and π/2, so both $\sin\frac u2$ and $\cos\frac u2$ will be positive. Then we find

$\cos\dfrac u2=\sqrt{\dfrac{1+\cos u}2}=\dfrac35$

$\sin\dfrac u2=\sqrt{\dfrac{1-\cos u}2}=\dfrac45$

and it follows that

$\tan\dfrac u2=\dfrac{\sin\frac u2}{\cos\frac u2}=\dfrac43$