Question

solve the following triangle.
a = 10°, b = 70°, a = 3
c ≈ 100°
(simplify your answer.)
b ≈ □
(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: 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 \(A = 10^{\circ}\), \(B=70^{\circ}\), \(C = 100^{\circ}\), and \(a = 3\).
To find \(b\), use \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
So, \(b=\frac{a\sin B}{\sin A}\).
Substitute the values: \(a = 3\), \(\sin A=\sin10^{\circ}\approx0.1736\), \(\sin B=\sin70^{\circ}\approx0.9397\).
\(b=\frac{3\times0.9397}{0.1736}\).

Step2: Calculate \(b\)

\(b=\frac{2.8191}{0.1736}\approx16.24\).

Step3: Find \(c\)

Use \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
\(c=\frac{a\sin C}{\sin A}\).
Substitute \(a = 3\), \(\sin A=\sin10^{\circ}\approx0.1736\), \(\sin C=\sin100^{\circ}\approx0.9848\).
\(c=\frac{3\times0.9848}{0.1736}\).

Step4: Calculate \(c\)

\(c=\frac{2.9544}{0.1736}\approx17.02\).

Answer:

\(b\approx16.24\), \(c\approx17.02\)