Question

a right triangle has side a opposite angle a, side b opposite angle b, and side c as the hypotenuse (opposite right angle c).
the length of one side and the measure of one angle are given. use a calculator to find the length of each remaining side to 4 decimal places.
b = 15, ∠b = 17°

Explanation:

Step1: Find angle \(A\)

In a right - triangle, \(A + B=90^{\circ}\). Given \(B = 17^{\circ}\), then \(A=90^{\circ}-B\).
\(A = 90^{\circ}-17^{\circ}=73^{\circ}\)

Step2: Find side \(a\)

We know that \(\tan B=\frac{b}{a}\). Rearranging for \(a\), we get \(a=\frac{b}{\tan B}\).
Since \(b = 15\) and \(B = 17^{\circ}\), then \(a=\frac{15}{\tan(17^{\circ})}\).
Using a calculator, \(\tan(17^{\circ})\approx0.3057\), so \(a=\frac{15}{0.3057}\approx49.0775\)

Step3: Find side \(c\)

We know that \(\sin B=\frac{b}{c}\). Rearranging for \(c\), we get \(c=\frac{b}{\sin B}\).
Since \(b = 15\) and \(B = 17^{\circ}\), then \(c=\frac{15}{\sin(17^{\circ})}\).
Using a calculator, \(\sin(17^{\circ})\approx0.2924\), so \(c=\frac{15}{0.2924}\approx51.2996\)

Answer:

\(a\approx49.0775\), \(c\approx51.2996\)