Question

solve the right triangle abc, with c = 90°. b = 68.2°, b = 103 in. a = 21.8° (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) a = □ in. (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) c = □ in. (simplify your answer. type a whole number.)

Explanation:

Step1: Recall trigonometric ratios

In a right triangle, \(\tan B=\frac{b}{a}\), so we can solve for \(a\) using \(a = \frac{b}{\tan B}\). Also, \(\sin B=\frac{b}{c}\), so we can solve for \(c\) using \(c=\frac{b}{\sin B}\).

Step2: Calculate \(a\)

We know \(B = 68.2^\circ\) and \(b = 103\) in.
\(\tan(68.2^\circ)\approx2.512\) (using calculator)
\(a=\frac{103}{\tan(68.2^\circ)}=\frac{103}{2.512}\approx41.0\)

Step3: Calculate \(c\)

\(\sin(68.2^\circ)\approx0.928\) (using calculator)
\(c = \frac{103}{\sin(68.2^\circ)}=\frac{103}{0.928}\approx111\)

Answer:

\(a\approx\boxed{41.0}\) in, \(c\approx\boxed{111}\) in