Question

solve the right triangle shown in the figure. a = 31.4, c = 53.9 a ≈ □° (round to the nearest tenth as needed.) b ≈ □° (round to the nearest tenth as needed.) b ≈ □ (round to the nearest hundredth as needed.)

Explanation:

Step1: Find side b using Pythagorean theorem

By the Pythagorean theorem $b=\sqrt{c^{2}-a^{2}}$. Substitute $a = 31.4$ and $c = 53.9$ into the formula:
$b=\sqrt{53.9^{2}-31.4^{2}}=\sqrt{(53.9 + 31.4)(53.9 - 31.4)}=\sqrt{85.3\times22.5}=\sqrt{1919.25}\approx43.81$

Step2: Find angle A using sine - function

We know that $\sin A=\frac{a}{c}$. So $A=\sin^{- 1}(\frac{a}{c})$. Substitute $a = 31.4$ and $c = 53.9$:
$A=\sin^{-1}(\frac{31.4}{53.9})\approx35.4^{\circ}$

Step3: Find angle B

Since the sum of angles in a triangle is $180^{\circ}$ and in a right - triangle $C = 90^{\circ}$, then $B=90^{\circ}-A$.
$B = 90^{\circ}-35.4^{\circ}=54.6^{\circ}$

Answer:

$A\approx35.4^{\circ}$
$B\approx54.6^{\circ}$
$b\approx43.81$