Question

solve the right triangle shown in the figure. a = 28.9, c = 54.2 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 $a = 28.9$ and $c = 54.2$:
$b=\sqrt{54.2^{2}-28.9^{2}}=\sqrt{(54.2 + 28.9)(54.2-28.9)}=\sqrt{83.1\times25.3}=\sqrt{2092.43}\approx45.74$

Step2: Find angle A using sine function

$\sin A=\frac{a}{c}$, so $A=\sin^{- 1}(\frac{a}{c})$. Substitute $a = 28.9$ and $c = 54.2$:
$A=\sin^{-1}(\frac{28.9}{54.2})\approx32.2^{\circ}$

Step3: Find angle B

Since the sum of angles in a triangle is $180^{\circ}$ and in a right - triangle $C = 90^{\circ}$, then $B=90^{\circ}-A$.
$B = 90^{\circ}-32.2^{\circ}=57.8^{\circ}$

Answer:

$A\approx32.2^{\circ}$
$B\approx57.8^{\circ}$
$b\approx45.74$