Question

Find the exact value of the trigonometric expression given that sin(u) = -(3/5),

Answer 1

Recall that

$\cot(u+v)=\dfrac{\cos(u+v)}{\sin(u+v)}=\dfrac{\cos u\cos v-\sin u\sin v}{\sin u\cos v+\cos u\sin v}=\dfrac{\cot u\cot v-1}{\cot v+\cot u}$

Also, recall that for all $\theta$,

$\cos^2\theta+\sin^2\theta=1$

With $\frac{3\pi}2$, we can expect $\cos u>0$, and with $0$, $\sin v>0$. So from the above identity it follows that

$\cos u=\sqrt{1-\sin^2u}=\dfrac45\implies\cot u=\dfrac{\cos u}{\sin u}=-\dfrac43$

and

$\sin v=\sqrt{1-\cos^2v}=\dfrac8{17}\implies\cot v=\dfrac{\cos v}{\sin v}=\dfrac{15}8$

and so

$\cot(u+v)=\dfrac{-\frac43\frac{15}8-1}{\frac{15}8-\frac43}=\boxed{-\dfrac{84}{13}}$