Question

determine the remaining sides and angles of the triangle abc.
a = 106.91°, c = 24.53°, c = 200

b = 48.56 °
a ≈ □
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. We know angle \(A = 106.91^\circ\), angle \(C=24.53^\circ\), and side \(c = 200\). We need to find side \(a\). So we use the ratio \(\frac{a}{\sin A}=\frac{c}{\sin C}\).

Step2: Solve for \(a\)

From \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we can re - arrange the formula to solve for \(a\): \(a=\frac{c\times\sin A}{\sin C}\)

Substitute the given values: \(c = 200\), \(A=106.91^\circ\), \(C = 24.53^\circ\)

First, calculate \(\sin(106.91^\circ)\) and \(\sin(24.53^\circ)\)

\(\sin(106.91^\circ)=\sin(180^\circ - 73.09^\circ)=\sin(73.09^\circ)\approx0.9563\)

\(\sin(24.53^\circ)\approx0.4145\)

Then, \(a=\frac{200\times\sin(106.91^\circ)}{\sin(24.53^\circ)}=\frac{200\times0.9563}{0.4145}\)

Calculate the numerator: \(200\times0.9563 = 191.26\)

Then, \(a=\frac{191.26}{0.4145}\approx461.4\) (rounded to the nearest tenth)

Answer:

\(461.4\)