Question

Let u and v be angles in the first quadrant, and let sinu=1/3 and sinv=1/4. Then cos⁡u = , cos⁡v = ,sin⁡(u+v) = .

Answer 1

Recall that $\sin^2x+\cos^2x=1$.

This means

$\cos u=\sqrt{1-\sin^2u}=\sqrt{1-\dfrac19}=\dfrac{\sqrt8}3=\dfrac{2\sqrt2}3$
$cos v=sqrt{1-\sin^2v}=\sqrt{1-\dfrac1{16}}=\dfrac{\sqrt{15}}4$
$\sin(u+v)=\sin u\cos v+\sin v\cos u=\dfrac13\times\dfrac{\sqrt{15}}4+\dfrac14\times\dfrac{2\sqrt2}3=\dfrac{2\sqrt2+\sqrt{15}}{12}$