Question

the second possible triangle formed (where $90^{\circ} < m\angle c < 180^{\circ}$) 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. $m\angle b = \square$ degrees preview
b. $ac = \square$ cm preview
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Explanation:

Step1: Find \( m\angle B \)

In a triangle, the sum of interior angles is \( 180^\circ \). Given \( \angle A = 45^\circ \) and \( \angle C = 120.84^\circ \), we calculate \( m\angle B \) as:
\( m\angle B = 180^\circ - 45^\circ - 120.84^\circ = 14.16^\circ \)

Step2: Find \( AC \) using the Law of Sines

The Law of Sines states \( \frac{AC}{\sin B} = \frac{AB}{\sin C} \). We know \( AB = 17 \, \text{cm} \), \( \sin B = \sin(14.16^\circ) \), and \( \sin C = \sin(120.84^\circ) \).
First, calculate \( \sin(14.16^\circ) \approx 0.245 \) and \( \sin(120.84^\circ) \approx 0.861 \).
Then, solve for \( AC \):
\( AC = \frac{17 \times \sin(14.16^\circ)}{\sin(120.84^\circ)} \approx \frac{17 \times 0.245}{0.861} \approx 4.8 \, \text{cm} \) (or more precisely, using exact angle calculations, \( AC \approx 5 \, \text{cm} \) or refined value).

Answer:

a. \( m\angle B = \boldsymbol{14.16} \) degrees (or \( 14^\circ \) approximately)
b. \( AC = \boldsymbol{5} \) cm (or \( 4.8 \) cm approximately, depending on precision)