Question

!!!PLEASE HELP ME !!! AND INCLUDE AN EXPLANATION

Answer 1

Since you have the 42-deg angle and the length of the opposite side, 31, you can use the Law of Sines.

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

We will use the law of sines first to find x.

Once we know x, we can find the angle opposite y.

Then we can find y.

$\dfrac{31}{\sin 42^\circ} = \dfrac{37}{\sin x}$

$\sin x = \dfrac{37\sin 42^\circ}{31}$

$\sin x = 0.798639$

$x = \sin^{-1} 0.798639 = 53^\circ$

Let z = the angle opposite y.

z + 42 + 53 = 180

z = 85

$\dfrac{31}{\sin 42^\circ} = \dfrac{y}{\sin 85^\circ}$

$y = \dfrac{31 \sin 85^\circ}{\sin 42^\circ}$

$y = 46.2$