Question

solve the right triangle shown in the figure.
a = 27.8, c = 52.3
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 $a = 27.8$ and $c = 52.3$:
$b=\sqrt{52.3^{2}-27.8^{2}}=\sqrt{(52.3 + 27.8)(52.3-27.8)}=\sqrt{80.1\times24.5}=\sqrt{1962.45}\approx44.30$

Step2: Find angle A using sine - function

$\sin(A)=\frac{a}{c}$. Substitute $a = 27.8$ and $c = 52.3$. Then $A=\sin^{-1}(\frac{27.8}{52.3})\approx32.4^{\circ}$

Step3: Find angle B

Since the sum of angles in a triangle is $180^{\circ}$ and $\angle C = 90^{\circ}$, then $B=90^{\circ}-A$. So $B = 90^{\circ}-32.4^{\circ}=57.6^{\circ}$

Answer:

$A\approx32.4^{\circ}$
$B\approx57.6^{\circ}$
$b\approx44.30$