Question

opp 12 cm
x hyp
50°
adj
sin50° = 12/x
x·sin50°/sin50 = 12/sin50
x = 15.7 cm
hyp
opp 2 m
10°
x
adj
tan10° = 2/x
x·tan10°/tan10 = 2/tan10

Explanation:

Step1: Identify trig - ratio for first triangle

Given an angle of $37^{\circ}$ and adjacent side of $8$ m, and we want the opposite side $x$. We use $\tan\theta=\frac{\text{opp}}{\text{adj}}$. So $\tan37^{\circ}=\frac{x}{8}$, and $x = 8\times\tan37^{\circ}\approx8\times0.7536 = 6.0$ m.

Step2: Identify trig - ratio for second triangle

Given an angle of $50^{\circ}$ and opposite side of $12$ cm, and we want the hypotenuse $x$. We use $\sin\theta=\frac{\text{opp}}{\text{hyp}}$. So $\sin50^{\circ}=\frac{12}{x}$, then $x=\frac{12}{\sin50^{\circ}}\approx\frac{12}{0.766}=15.7$ cm.

Step3: Identify trig - ratio for third triangle

Given an angle of $10^{\circ}$ and opposite side of $2$ m, and we want the adjacent side $x$. We use $\tan\theta=\frac{\text{opp}}{\text{adj}}$. So $\tan10^{\circ}=\frac{2}{x}$, then $x=\frac{2}{\tan10^{\circ}}\approx\frac{2}{0.1763}=11.3$ m.

Answer:

For the first triangle, $x = 6.0$ m; for the second triangle, $x = 15.7$ cm; for the third triangle, $x\approx11.3$ m.