Question

solve the triangle.
write each answer as an integer or as a decimal rounded to the nearest tenth.

Explanation:

Step1: Find side $FG$ via Law of Cosines

$$ FG^2 = 14^2 + 12^2 - 2(14)(12)\cos(20^\circ) $$

$$ FG^2 = 196 + 144 - 336\cos(20^\circ) $$

$$ FG^2 = 340 - 336(0.9397) $$

$$ FG^2 = 340 - 315.7392 = 24.2608 $$

$$ FG = \sqrt{24.2608} \approx 4.9 $$

Step2: Find $\angle F$ via Law of Sines

$$ \frac{\sin\angle F}{12} = \frac{\sin(20^\circ)}{4.9} $$

$$ \sin\angle F = \frac{12\sin(20^\circ)}{4.9} $$

$$ \sin\angle F = \frac{12(0.3420)}{4.9} = \frac{4.104}{4.9} \approx 0.8376 $$

$$ \angle F = \arcsin(0.8376) \approx 56.9^\circ $$

Step3: Find $\angle G$ via angle sum

$$ \angle G = 180^\circ - 20^\circ - 56.9^\circ $$

$$ \angle G = 103.1^\circ $$

Answer:

$FG \approx 4.9$, $\angle F \approx 56.9^\circ$, $\angle G \approx 103.1^\circ$