Question

1 using law of sines to
use the law of sines to solve for the indicated values. round decimal
answers to the nearest tenth.

a) ( mangle c=) type your answer...

b) ( b=) type your answer... ( cm)

c) ( c=) type your answer... ( cm)

Explanation:

Step1: Find \(m\angle C\)

The sum of angles in a triangle is \(180^{\circ}\). So \(m\angle C=180^{\circ}-(32^{\circ} + 73^{\circ})\)

$$m\angle C=180^{\circ}-105^{\circ}=75^{\circ}$$

Step2: Use the Law of Sines to find \(b\)

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(a = 23\) (opposite \(\angle A = 32^{\circ}\)), \(\angle B=73^{\circ}\), \(\angle C = 75^{\circ}\). To find \(b\) (opposite \(\angle B\)):
\(\frac{b}{\sin73^{\circ}}=\frac{23}{\sin32^{\circ}}\)

$$b=\frac{23\times\sin73^{\circ}}{\sin32^{\circ}}$$

\(\sin73^{\circ}\approx0.956\), \(\sin32^{\circ}\approx0.5299\)

$$b=\frac{23\times0.956}{0.5299}=\frac{21.988}{0.5299}\approx41.5$$

Step3: Use the Law of Sines to find \(c\)

To find \(c\) (opposite \(\angle C\)): \(\frac{c}{\sin75^{\circ}}=\frac{23}{\sin32^{\circ}}\)
\(\sin75^{\circ}\approx0.9659\)

$$c=\frac{23\times\sin75^{\circ}}{\sin32^{\circ}}=\frac{23\times0.9659}{0.5299}=\frac{22.2157}{0.5299}\approx41.9$$

Answer:

a) \(75\)
b) \(41.5\)
c) \(41.9\)