Answer
w = 18
x = 31.18
y = 18
z = 25.46
Work Step by Step
To complete this question, we are looking to find variables: x, y, z, & w.
We should start by examining what we know from the geometry. We know that there are two right angle triangles, meaning that at least one of the angles is 90$^{\circ}$.
With this information, we can separate the two triangles and then we have a 30-60-90 triangle and a 45-45-90 triangle, since all the angles in a triangle must add to 180$^{\circ}$.
Now, lets find w.
SOH = $\sin$ $\theta$ = $\frac{Opposite}{Hypotenuse}$
$\theta$ = 30$^{\circ}$ , Opposite = w , Hypotenuse = 36
$\sin$ 30$^{\circ}$ = $\frac{w}{36}$
36$\times\sin$ 30$^{\circ}$ = w
w = 18
We can do the same to find x.
CAH = $\cos$ $\theta$ = $\frac{Adjacent}{Hypotenuse}$
$\theta$ = 30$^{\circ}$ , Adjacent = x , Hypotenuse = 36
$\cos$ 30$^{\circ}$ = $\frac{x}{36}$
36$\times\cos$ 30$^{\circ}$ = x
x = 31.18
We can now use w to find y & z.
TOA = $\tan\theta$ = $\frac{Opposite}{Adjacent}$
$\theta$ = 45$^{\circ}$ , w = 18
$\tan\theta$ = $\frac{w}{y}$
$\tan45^{\circ}$ = $\frac{18}{y}$
y = $\frac{18}{\tan45^{\circ}}$
y = 18
And, lastly.
$\sin$ $\theta$ = $\frac{w}{z}$
$\sin$ $45$$^{\circ}$ = $\frac{18}{z}$
z = $\frac{18}{\sin 45^{\circ}}$
z = 25.46
