One of the $\sin{x}$, $\cos{x}$ or $\tan{x}$ is given. If $x$ is in the specified range, find the other two.

Question

$\sin{x}=\frac{3}{5}, x\in[\frac{\pi}{2}, \pi]$

I know that if $\sin{x}=\frac{3}{5}$ then the length of the opposite edge is 3 and length of the hypotenuse is 5. So, that,

$\cos{x}=\frac{4}{5}, \tan{x}=\frac{3}{4}$

My question is about, how we can check that given x radian is in the specified range?

Thanks.

Answer 1

$x\in[\frac{\pi}2,\pi]$ means the angle is in Quadrant $2$, where sine ratios are positive, but cosine and tangent ratios are negative -- you probably learned this as the CAST rule.

In this case, your answers should be $\cos x=-\frac45$ and $\tan x=-\frac34$.