Question

solve the right triangle abc, where c = 90°. give angles in degrees and minutes.

a = 13.00 m, c = 28.00 m

b ≈ \\(\square\\) m (round to the nearest hundredth as needed.)
a = \\(\square\\)°\\(\square\\)′ (round to the nearest minute as needed.)
b = \\(\square\\)°\\(\square\\)′ (round to the nearest minute as needed.)

Explanation:

Step1: Find side \( b \) using Pythagorean theorem

In a right triangle, \( a^2 + b^2 = c^2 \). So, \( b = \sqrt{c^2 - a^2} \).
Given \( a = 13.00 \), \( c = 28.00 \), then \( b = \sqrt{28.00^2 - 13.00^2} = \sqrt{784 - 169} = \sqrt{615} \approx 24.80 \)

Step2: Find angle \( A \) using sine function

\( \sin A = \frac{a}{c} = \frac{13.00}{28.00} \approx 0.4643 \)
Then \( A = \arcsin(0.4643) \approx 27.66^\circ \)
Convert decimal degrees to minutes: \( 0.66^\circ \times 60 \approx 40' \), so \( A \approx 27^\circ 40' \)

Step3: Find angle \( B \)

Since \( A + B = 90^\circ \), \( B = 90^\circ - A \approx 90^\circ - 27^\circ 40' = 62^\circ 20' \)

Answer:

\( b \approx \boldsymbol{24.80} \) m
\( A = \boldsymbol{27^\circ 40'} \)
\( B = \boldsymbol{62^\circ 20'} \)