Question

solve the triangle.
c = 86°
(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 \(a = 107\), \(A=56^{\circ}\), \(B = 38^{\circ}\), \(C=86^{\circ}\).

Step2: Find \(b\)

From \(\frac{a}{\sin A}=\frac{b}{\sin B}\), we can solve for \(b\). Substitute the values: \(b=\frac{a\sin B}{\sin A}\).

$$ b=\frac{107\times\sin38^{\circ}}{\sin56^{\circ}} $$

$$ \sin38^{\circ}\approx0.6157,\sin56^{\circ}\approx0.8290 $$

$$ b=\frac{107\times0.6157}{0.8290}\approx79.6 $$

Step3: Find \(c\)

From \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we can solve for \(c\). Substitute the values: \(c=\frac{a\sin C}{\sin A}\).

$$ c=\frac{107\times\sin86^{\circ}}{\sin56^{\circ}} $$

$$ \sin86^{\circ}\approx0.9976,\sin56^{\circ}\approx0.8290 $$

$$ c=\frac{107\times0.9976}{0.8290}\approx128.1 $$

Answer:

\(b\approx79.6\), \(c\approx128.1\)