Question

solve the triangle.
a = 63°, b = 47°, a = 10
c = 70°
(do not round until the final answer. then round to the nearest degree as needed.)
b ≈
(do not round until the final answer. then round to the nearest tenth as needed.)
c ≈
(do not round until the final answer. then round to the nearest tenth 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 \(A = 63^{\circ}\), \(B=47^{\circ}\), \(C = 70^{\circ}\), and \(a = 10\).
To find \(b\), we use \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
So, \(b=\frac{a\sin B}{\sin A}\).
Substitute the values: \(b=\frac{10\times\sin(47^{\circ})}{\sin(63^{\circ})}\).
Since \(\sin(47^{\circ})\approx0.731\) and \(\sin(63^{\circ})\approx0.891\), then \(b=\frac{10\times0.731}{0.891}\approx8.2\).

Step2: Find \(c\) using the Law of Sines

Using \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we have \(c=\frac{a\sin C}{\sin A}\).
Substitute \(a = 10\), \(\sin(63^{\circ})\approx0.891\), \(\sin(70^{\circ})\approx0.940\).
So, \(c=\frac{10\times0.940}{0.891}\approx10.6\).

Answer:

\(b\approx8.2\), \(c\approx10.6\)