Question

determine the remaining sides and angles of the triangle abc.
a = 18.47° b = 20.32° c = 19.44 m
c = □°
a ≈ □ m
(round to the nearest hundredth as needed.)
b ≈ □ m
(round to the nearest hundredth as needed.)

Explanation:

Step1: Find angle C

The sum of angles in a triangle is \(180^\circ\). So, \(C = 180^\circ - A - B\).
\(C = 180^\circ - 18.47^\circ - 20.32^\circ = 141.21^\circ\)

Step2: Use the Law of Sines to find side a

The Law of Sines states that \(\frac{a}{\sin A}=\frac{c}{\sin C}\). So, \(a=\frac{c\sin A}{\sin C}\).
First, calculate \(\sin A=\sin(18.47^\circ)\approx0.316\), \(\sin C=\sin(141.21^\circ)=\sin(180^\circ - 38.79^\circ)=\sin(38.79^\circ)\approx0.627\) (since \(\sin(180^\circ - x)=\sin x\)).
Then, \(a=\frac{19.44\times0.316}{0.627}\approx\frac{6.143}{0.627}\approx9.80\) m

Step3: Use the Law of Sines to find side b

Using the Law of Sines \(\frac{b}{\sin B}=\frac{c}{\sin C}\). So, \(b=\frac{c\sin B}{\sin C}\).
Calculate \(\sin B=\sin(20.32^\circ)\approx0.347\).
Then, \(b=\frac{19.44\times0.347}{0.627}\approx\frac{6.746}{0.627}\approx10.76\) m

Answer:

\(C = 141.21^\circ\)
\(a\approx9.80\) m
\(b\approx10.76\) m