Question

a right triangle has side lengths ac = 7 inches, bc = 24 inches, and ab = 25 inches. what are the measures of the angles in triangle abc? m∠a ≈ 46.2°, m∠b ≈ 43.8°, m∠c = 90°; m∠a ≈ 73.0°, m∠b ≈ 17.0°, m∠c = 90°; m∠a ≈ 73.7°, m∠b ≈ 16.3°, m∠c = 90°; m∠a ≈ 74.4°, m∠b ≈ 15.6°, m∠c = 90°

Explanation:

Step1: Use trigonometric ratios

In a right - triangle \(ABC\) with \(\angle C = 90^{\circ}\), \(\sin A=\frac{BC}{AB}\) and \(\sin B=\frac{AC}{AB}\).
Given \(AC = 7\), \(BC = 24\), \(AB = 25\).
For \(\angle A\), \(\sin A=\frac{BC}{AB}=\frac{24}{25}=0.96\).
Using the inverse - sine function \(A=\sin^{- 1}(0.96)\).
For \(\angle B\), \(\sin B=\frac{AC}{AB}=\frac{7}{25}=0.28\).
Using the inverse - sine function \(B=\sin^{-1}(0.28)\).

Step2: Calculate the angles

We know that \(A=\sin^{-1}(0.96)\approx73.7^{\circ}\) (since \(\sin(73.7^{\circ})\approx0.96\)).
And \(B=\sin^{-1}(0.28)\approx16.3^{\circ}\) (since \(\sin(16.3^{\circ})\approx0.28\)).
Also, since it is a right - triangle \(\angle C = 90^{\circ}\).

Answer:

\(m\angle A\approx73.7^{\circ},m\angle B\approx16.3^{\circ},m\angle C = 90^{\circ}\)