Question

use the right triangle and the given information to solve the triangle.
b = 10, b = 35°; find a, c, and a
a = (round to the nearest hundredth as needed.)

Explanation:

Step1: Find angle \(A\)

In a right - triangle, \(A + B=90^{\circ}\). Given \(B = 35^{\circ}\), then \(A=90^{\circ}-B\).
\(A = 90^{\circ}-35^{\circ}=55^{\circ}\)

Step2: Find side \(a\)

We know that \(\tan B=\frac{b}{a}\) (since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) in a right - triangle). Given \(b = 10\) and \(B = 35^{\circ}\), then \(a=\frac{b}{\tan B}\).
\(\tan35^{\circ}\approx0.7002\), so \(a=\frac{10}{\tan35^{\circ}}=\frac{10}{0.7002}\approx14.28\)

Step3: Find side \(c\)

We can use the sine function \(\sin B=\frac{b}{c}\) (since \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)). Then \(c=\frac{b}{\sin B}\).
\(\sin35^{\circ}\approx0.5736\), so \(c=\frac{10}{\sin35^{\circ}}=\frac{10}{0.5736}\approx17.43\)

Answer:

\(a\approx14.28\), \(c\approx17.43\), \(A = 55^{\circ}\)