Question

find fg.
write your answer as an integer or as a decimal rounded to the nearest tenth.
fg =
submit

Explanation:

Step1: Identify the triangle type and trigonometric relation

This is a right - triangle \( \triangle FGH \) with \( \angle G = 90^{\circ} \), \( \angle H=37^{\circ} \) and hypotenuse \( HF=\sqrt{38} \). We want to find the length of \( FG \), which is the side opposite to \( \angle H \). The sine of an angle in a right - triangle is defined as \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \). So, \( \sin(\angle H)=\frac{FG}{HF} \).

Step2: Substitute the known values

We know that \( \angle H = 37^{\circ} \), \( HF=\sqrt{38}\approx6.164 \) (since \( \sqrt{36} = 6 \) and \( \sqrt{49}=7 \), \( \sqrt{38}\approx6.164 \)) and \( \sin(37^{\circ})\approx0.602 \). From \( \sin(37^{\circ})=\frac{FG}{\sqrt{38}} \), we can solve for \( FG \) by multiplying both sides of the equation by \( \sqrt{38} \). So, \( FG=\sin(37^{\circ})\times\sqrt{38} \).

Step3: Calculate the value of \( FG \)

Substitute \( \sin(37^{\circ})\approx0.602 \) and \( \sqrt{38}\approx6.164 \) into the formula: \( FG\approx0.602\times6.164 \approx 3.7 \) (rounded to the nearest tenth).

Answer:

\( 3.7 \)