Question

solve the following triangle.

b = 10°, c = 60°, b = 6

a ≈ □°
(simplify your answer.)
a ≈ □
(type an integer or decimal rounded to two decimal places as needed.)
c ≈ □
(type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Find angle \(A\)

The sum of angles in a triangle is \(180^{\circ}\). So \(A = 180^{\circ}-B - C\).
Substitute \(B = 10^{\circ}\) and \(C=60^{\circ}\):
\(A=180^{\circ}-10^{\circ}-60^{\circ}=110^{\circ}\)

Step2: Use the Law of Sines to find side \(a\)

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
We know \(b = 6\), \(B = 10^{\circ}\), \(A = 110^{\circ}\).
So \(a=\frac{b\sin A}{\sin B}\)
\(\sin A=\sin(110^{\circ})\approx0.9397\), \(\sin B=\sin(10^{\circ})\approx0.1736\)
\(a=\frac{6\times0.9397}{0.1736}\approx32.48\)

Step3: Use the Law of Sines to find side \(c\)

Using the Law of Sines \(\frac{c}{\sin C}=\frac{b}{\sin B}\)
\(\sin C=\sin(60^{\circ})\approx0.8660\), \(b = 6\), \(\sin B\approx0.1736\)
\(c=\frac{b\sin C}{\sin B}=\frac{6\times0.8660}{0.1736}\approx29.85\)

Answer:

\(A\approx110^{\circ}\), \(a\approx32.48\), \(c\approx29.85\)