Question

let’s begin by using the law of sines to determine the missing information in the case where \\( \angle c \\) is acute. this case is more straightforward and works just like the examples from the beginning of this investigation. the first possible triangle formed (where \\( 0^{\circ} < m\angle c < 90^{\circ} \\)) is shown below. use this triangle to complete the following questions. note that this question allows you to use trig functions that evaluate for angle measures in degrees. to remind yourself how to evaluate trig functions for angle measures in degrees in this program, click here. a. \\( m\angle c = \\) \\( \square \\) degrees preview b. \\( m\angle b = \\) \\( \square \\) degrees preview c. \\( ac = \\) \\( \square \\) cm preview submit

Explanation:

Step1: Apply Law of Sines for ∠C

In triangle \(ABC\), \(\angle A = 45^\circ\), \(AB = 17\) cm, \(BC = 14\) cm. By Law of Sines: \(\frac{\sin C}{AB}=\frac{\sin A}{BC}\). So \(\sin C=\frac{AB\sin A}{BC}=\frac{17\sin 45^\circ}{14}\). Calculate \(\sin 45^\circ=\frac{\sqrt{2}}{2}\approx0.7071\), then \(\sin C=\frac{17\times0.7071}{14}\approx\frac{12.0207}{14}\approx0.8586\). So \(m\angle C=\arcsin(0.8586)\approx59.1^\circ\) (since \(C\) is acute).

Step2: Find \(m\angle B\)

Sum of angles in triangle is \(180^\circ\). So \(m\angle B = 180^\circ - 45^\circ - 59.1^\circ = 75.9^\circ\).

Step3: Apply Law of Sines for \(AC\)

Using Law of Sines: \(\frac{AC}{\sin B}=\frac{BC}{\sin A}\). So \(AC=\frac{BC\sin B}{\sin A}=\frac{14\sin 75.9^\circ}{\sin 45^\circ}\). \(\sin 75.9^\circ\approx0.969\), \(\sin 45^\circ\approx0.7071\). Then \(AC=\frac{14\times0.969}{0.7071}\approx\frac{13.566}{0.7071}\approx19.2\) cm.

Answer:

a. \(m\angle C\approx\boldsymbol{59.1}\) degrees
b. \(m\angle B\approx\boldsymbol{75.9}\) degrees
c. \(AC\approx\boldsymbol{19.2}\) cm