Question

for triangle abc, we are given that m∠b = 141.1°, ab = 15.1 cm, and ac = 32.6 cm. we can use the law of sines, as well as other properties of triangles, to determine all of the missing information. 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. determine the missing information for this triangle. a. m∠c = select an answer preview b. m∠a = select an answer preview c. bc = cm select an answer preview submit question 7. points possible: 4 unlimited attempts. message instructor about this question post this question to forum

Explanation:

Step1: Apply Law of Sines for \( m\angle C \)

In \( \triangle ABC \), Law of Sines: \( \frac{AC}{\sin B}=\frac{AB}{\sin C} \). Given \( AC = 32.6 \, \text{cm} \), \( AB = 15.1 \, \text{cm} \), \( m\angle B = 141.1^\circ \). So \( \sin C=\frac{AB \cdot \sin B}{AC}=\frac{15.1 \cdot \sin(141.1^\circ)}{32.6} \). Calculate \( \sin(141.1^\circ)=\sin(180^\circ - 38.9^\circ)=\sin(38.9^\circ)\approx0.629 \). Then \( \sin C\approx\frac{15.1 \cdot 0.629}{32.6}\approx\frac{9.4979}{32.6}\approx0.291 \). So \( m\angle C\approx\arcsin(0.291)\approx16.9^\circ \).

Step2: Find \( m\angle A \)

Triangle angle sum: \( m\angle A + m\angle B + m\angle C = 180^\circ \). So \( m\angle A = 180^\circ - 141.1^\circ - 16.9^\circ = 22^\circ \).

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

Law of Sines: \( \frac{BC}{\sin A}=\frac{AC}{\sin B} \). So \( BC=\frac{AC \cdot \sin A}{\sin B}=\frac{32.6 \cdot \sin(22^\circ)}{\sin(141.1^\circ)} \). \( \sin(22^\circ)\approx0.375 \), \( \sin(141.1^\circ)\approx0.629 \). Then \( BC\approx\frac{32.6 \cdot 0.375}{0.629}\approx\frac{12.225}{0.629}\approx19.44 \, \text{cm} \).

Answer:

a. \( m\angle C \approx 16.9^\circ \)
b. \( m\angle A = 22^\circ \)
c. \( BC \approx 19.44 \, \text{cm} \)