Question

solve the right triangle abc, where c = 90°. a = 76.3 yd, b = 41.7 yd c = □ yd (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) a = □°□ (simplify your answers. type integers. round to the nearest ten minutes as needed.) b = □°□ (simplify your answers. type integers. round to the nearest ten minutes as needed.)

Explanation:

Step1: Find side c using Pythagorean theorem

By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 76.3\) and \(b=41.7\). So \(c=\sqrt{(76.3)^{2}+(41.7)^{2}}=\sqrt{5821.69 + 1738.89}=\sqrt{7560.58}\approx87.0\)

Step2: Find angle A using tangent function

\(\tan A=\frac{a}{b}\), so \(\tan A=\frac{76.3}{41.7}\approx1.83\). Then \(A=\arctan(1.83)\approx61.4^{\circ}\). Convert the decimal part to minutes: \(0.4\times60 = 24\), so \(A = 61^{\circ}24'\)

Step3: Find angle B

Since \(A + B=90^{\circ}\) in a right - triangle, \(B = 90^{\circ}-A\). \(B=90^{\circ}- 61^{\circ}24'=28^{\circ}36'\)

Answer:

c = 87.0 yd
A = 61°24'
B = 28°36'