Question

in a tournament, a professional golfer knows that she is 200 yards from the hole. a spectator is watching her play and is 140 yards away from the golfer.

if the spectator has an angle of 110° between the golfer and the hole, what is the angle that the golfer has between the spectator and the hole?
70.0°
41.1°
28.9°
10.0°

Explanation:

Step1: Assign labels to triangle parts

Let:

• $A$ = Golfer's position, angle to find = $\angle A$
• $B$ = Spectator's position, $\angle B = 110^\circ$
• $C$ = Hole's position
• Side opposite $A$: $a = 200$ yd
• Side opposite $B$: $b = 140$ yd

Step2: Apply Law of Sines

Law of Sines: $\frac{\sin A}{a} = \frac{\sin B}{b}$
Rearrange to solve for $\sin A$:
$\sin A = \frac{a \cdot \sin B}{b}$
Substitute values:
$\sin A = \frac{200 \cdot \sin(110^\circ)}{140}$

Step3: Calculate $\sin A$

First, $\sin(110^\circ) \approx 0.9397$
$\sin A \approx \frac{200 \cdot 0.9397}{140} \approx \frac{187.94}{140} \approx 1.3424$
Note: This is impossible, correct side labeling: side opposite $\angle B$ is 200 yd, side opposite $\angle A$ is 140 yd
Correct formula: $\sin A = \frac{140 \cdot \sin(110^\circ)}{200}$
$\sin A \approx \frac{140 \cdot 0.9397}{200} \approx \frac{131.558}{200} \approx 0.6578$

Step4: Find $\angle A$

$\angle A = \arcsin(0.6578) \approx 41.1^\circ$

Answer:

41.1°