Question

solve each of the following right triangles for all missing angle sizes and all missing side lengths in problems 26 to 36. all of the triangles correspond to the standard right triangle, abc, shown below. round answers to the nearest tenth. 26. ∠a = 29°, a = 32 27. ∠b = 68.3°, a = 417 28. ∠a = 47.7°, c = 132.2 29. b = 731, c = 818 30. ∠b = 32.4°, a = 27.5 31. a = 2.2, c = 3.2 32. ∠b = 52.6°, b = 1.8 33. a = 16.1, b = 34.2 34. ∠a = 38°, b = 8.0 35. ∠b = 43.8, c = 6.3 36. b = 221, c = 247

Explanation:

Step1: Find angle B

In a right - triangle \(ABC\) with \(\angle C = 90^{\circ}\), we know that \(\angle A+\angle B = 90^{\circ}\). Given \(\angle A=29^{\circ}\), then \(\angle B=90^{\circ}-\angle A = 90^{\circ}- 29^{\circ}=61^{\circ}\).

Step2: Find side b

We use the tangent function \(\tan A=\frac{a}{b}\). So \(b = \frac{a}{\tan A}\). Substituting \(a = 32\) and \(\angle A=29^{\circ}\), we have \(b=\frac{32}{\tan29^{\circ}}\approx\frac{32}{0.5543}\approx57.7\).

Step3: Find side c

We use the sine function \(\sin A=\frac{a}{c}\). So \(c=\frac{a}{\sin A}\). Substituting \(a = 32\) and \(\angle A = 29^{\circ}\), we get \(c=\frac{32}{\sin29^{\circ}}\approx\frac{32}{0.4848}\approx66.0\).

Answer:

\(\angle B = 61^{\circ}\), \(b\approx57.7\), \(c\approx66.0\)