Question

solve the right triangle abc, where c = 90°. give angles in degrees and minutes.
a = 13.00 m, c = 23.00 m
b ≈ m (round to the nearest hundredth as needed.)
a = ° (round to the nearest minute as needed.)
b = ° (round to the nearest minute as needed.)

Explanation:

Step1: Find side b using Pythagorean theorem

By the Pythagorean theorem $b=\sqrt{c^{2}-a^{2}}$. Substitute $a = 13.00$ and $c = 23.00$ into the formula:
$b=\sqrt{23.00^{2}-13.00^{2}}=\sqrt{(23 + 13)(23 - 13)}=\sqrt{36\times10}=\sqrt{360}\approx18.97$ m.

Step2: Find angle A using sine function

$\sin A=\frac{a}{c}$. Substitute $a = 13.00$ and $c = 23.00$, then $A=\sin^{-1}(\frac{13}{23})$.
$A=\sin^{-1}(\frac{13}{23})\approx34.1^{\circ}$. Convert the decimal part to minutes: $0.1\times60 = 6'$. So $A = 34^{\circ}6'$.

Step3: Find angle B

Since the sum of angles in a triangle is $180^{\circ}$ and $C = 90^{\circ}$, then $B=90^{\circ}-A$.
$B = 90^{\circ}-34^{\circ}6'=55^{\circ}54'$.

Answer:

b $\approx$ 18.97 m
A = 34$^{\circ}$6$'$
B = 55$^{\circ}$54$'$