Question

solve the following triangle.
a = 50°, b = 50°, c = 9
c ≈ 80°
(simplify your answer.)
a ≈
(type an integer or decimal rounded to two decimal places as needed.)
b ≈
(type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).
We know \(A = 50^{\circ}\), \(B = 50^{\circ}\), \(C\approx80^{\circ}\), and \(c = 9\).
For side \(a\):
\(\frac{a}{\sin A}=\frac{c}{\sin C}\)
\(a=\frac{c\sin A}{\sin C}\)
Substitute the values: \(a=\frac{9\sin50^{\circ}}{\sin80^{\circ}}\)
Since \(\sin50^{\circ}\approx0.7660\) and \(\sin80^{\circ}\approx0.9848\)
\(a=\frac{9\times0.7660}{0.9848}\)
\(a=\frac{6.894}{0.9848}\approx6.91\)

Step2: For side \(b\)

Since \(A = B = 50^{\circ}\), by the Law of Sines (or because in a triangle equal angles have equal opposite sides)
\(\frac{b}{\sin B}=\frac{c}{\sin C}\)
\(b=\frac{c\sin B}{\sin C}\)
Since \(A = B\), \(\sin A=\sin B\)
So \(b = a\approx6.91\)

Answer:

\(a\approx6.91\), \(b\approx6.91\)