Question

question 5 or 10
in triangle abc, ( mangle a = 58^{circ} ), ( mangle b = 25^{circ} ), and ( b = 21 ). use the law of sines to find c. round your answer to the nearest tenth.
a. 49.3
b. 97.0
c. 10.5
d. 42.1

Explanation:

Step1: Find the measure of angle \(C\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(m\angle C=180^{\circ}-m\angle A - m\angle B\).
Substituting \(m\angle A = 58^{\circ}\) and \(m\angle B=25^{\circ}\), we get \(m\angle C=180^{\circ}-58^{\circ}-25^{\circ}=97^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(b = 21\), \(m\angle B = 25^{\circ}\), and \(m\angle C=97^{\circ}\).
From \(\frac{b}{\sin B}=\frac{c}{\sin C}\), we can solve for \(c\). Rearranging gives \(c=\frac{b\sin C}{\sin B}\).
Substitute \(b = 21\), \(\sin B=\sin(25^{\circ})\approx0.4226\), \(\sin C=\sin(97^{\circ})\approx0.9925\) into the formula:
\(c=\frac{21\times0.9925}{0.4226}\)
\(c=\frac{20.8425}{0.4226}\approx49.3\)

Answer:

A. 49.3