Question

determine the remaining sides and angles of the triangle abc.
a = 112.33°, c = 20.46°, c = 120

b = 47.21°
a ≈ \square
(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 \(A = 112.33^\circ\), angle \(C=20.46^\circ\), and side \(c = 120\). We want to find side \(a\). From the Law of Sines, we can set up the proportion \(\frac{a}{\sin A}=\frac{c}{\sin C}\).

Step2: Solve for \(a\)

Rearranging the proportion to solve for \(a\), we get \(a=\frac{c\times\sin A}{\sin C}\). Substitute the known values: \(c = 120\), \(\sin A=\sin(112.33^\circ)\), and \(\sin C=\sin(20.46^\circ)\).

First, calculate \(\sin(112.33^\circ)\approx\sin(180^\circ - 67.67^\circ)=\sin(67.67^\circ)\approx0.925\) (more accurately, using a calculator: \(\sin(112.33^\circ)\approx\sin(112.33)\approx0.9254\)). \(\sin(20.46^\circ)\approx0.349\) (more accurately, \(\sin(20.46^\circ)\approx0.3493\)).

Now, substitute into the formula: \(a=\frac{120\times\sin(112.33^\circ)}{\sin(20.46^\circ)}\).

Calculate the numerator: \(120\times\sin(112.33^\circ)\approx120\times0.9254 = 111.048\).

Calculate the denominator: \(\sin(20.46^\circ)\approx0.3493\).

Then, \(a=\frac{111.048}{0.3493}\approx317.9\) (after rounding to the nearest tenth).

Answer:

\(317.9\)