I would really appreciate if someone could show me how to solve this question. Thanks.1. Given that $\displaystyle Sin(x) = Cos({3 \pi}{/11})$, and x lies in the second quadrant, determine the measure of x.
Cosine of an angle = sine of its complement
$\displaystyle \cos \frac{3 \pi}{11}=\sin \left( \frac{\pi}{2}-\frac{3 \pi}{11}\right)=\sin \frac{5 \pi}{22}$
That's a Q1 angle and you want the angle in QII.
Sin is positive in Q1 and QII, so find the reference angle in QII.
$\displaystyle x=\pi-\frac{5 \pi}{22}=\frac{17 \pi}{22}$
well, if you're a glutton for punishment, you can always do
$\displaystyle \tan \frac x2 = \frac {\sin \frac x2}{\cos \frac x2}$
now, each of $\displaystyle \sin \frac x2$ and $\displaystyle \cos \frac x2$ can be derived using the double angle formula for $\displaystyle \cos 2x$
or even easier: recall that $\displaystyle \tan 2x = \frac {2 \tan x}{1 - \tan^2 x}$. now replace $\displaystyle x$ with $\displaystyle \frac x2$ everywhere and solve for $\displaystyle \tan \frac x2$
So what if you didn't cover it? Be ahead of your class and derive the half-angle identities. And you can do it using your double angle identities. Attempt it for abit, and if you reach to it or hit a dead-end, you can go here:
http://math.ucsd.edu/~wgarner/math4c.../halfangle.htm
Using the double angle identity for tan to find tan(x/2) is just complicated and prone to error.