Question

law of sines
what is the approximate measure of angle f? use the law of sines to find the answer.
11.5°
44.4°
68.0°
(diagram: right triangle fgh with right angle at g, gh = 28, fh = 40)

Explanation:

Step1: Recall Law of Sines

The Law of Sines states that in any triangle, $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$, where $A, B, C$ are angles and $a, b, c$ are the lengths of the sides opposite those angles. In right triangle $FGH$ (right - angled at $G$), we know side $GH = 28$ (opposite angle $F$), hypotenuse $FH = 40$ (opposite angle $G = 90^{\circ}$). Let $\angle F=\theta$. Then by Law of Sines, $\frac{\sin\theta}{GH}=\frac{\sin G}{FH}$.

Step2: Substitute values

We know $GH = 28$, $FH = 40$, and $\sin G=\sin90^{\circ}=1$. Substituting these values into the Law of Sines formula: $\frac{\sin\theta}{28}=\frac{1}{40}$.

Step3: Solve for $\sin\theta$

Cross - multiply to get $\sin\theta=\frac{28}{40}=0.7$.

Step4: Find $\theta$

To find $\theta$, we take the inverse sine (arcsin) of $0.7$. So $\theta=\arcsin(0.7)\approx44.4^{\circ}$.

Answer:

$44.4^{\circ}$ (corresponding to the option with $44.4^{\circ}$)