Question

for triangle abc, we are given that m∠b = 141.8°, ab = 14.9 cm, and ac = 35.2 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 = 152 degrees preview b m∠a = 23 degrees preview c bc = 22.1 cm preview submit question 5 points possible: 6 unlimited attempts. score on last attempt: 5. score in gradebook: 5 message instructor about this question post this question to forum

Explanation:

Step1: Recall Law of Sines

The Law of Sines states that for a triangle \(ABC\) with sides \(a, b, c\) opposite angles \(A, B, C\) respectively, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Here, \(AB = c = 14.9\) cm (opposite \(\angle C\)), \(AC = b = 35.2\) cm (opposite \(\angle B\)), \(BC = a\) (opposite \(\angle A\)), and \(\angle B = 141.8^\circ\).

Step2: Find \(\angle C\) using Law of Sines

We know \(\frac{AC}{\sin B}=\frac{AB}{\sin C}\), so \(\sin C=\frac{AB\sin B}{AC}\). Substitute \(AB = 14.9\), \(AC = 35.2\), \(\sin B=\sin(141.8^\circ)\). \(\sin(141.8^\circ)=\sin(180^\circ - 38.2^\circ)=\sin(38.2^\circ)\approx0.618\). Then \(\sin C=\frac{14.9\times0.618}{35.2}\approx\frac{9.21}{35.2}\approx0.262\). So \(\angle C\approx\arcsin(0.262)\approx15.2^\circ\) (wait, maybe miscalculation earlier, let's recalculate: \(\sin(141.8^\circ)=\sin(141.8)\approx\sin(141.8)= \sin(180 - 38.2)= \sin(38.2)\approx0.618\). Then \(14.9\times0.618 = 14.9\times0.6 + 14.9\times0.018 = 8.94 + 0.2682 = 9.2082\). Then \(9.2082\div35.2\approx0.2616\). \(\arcsin(0.2616)\approx15.2^\circ\), so \(m\angle C\approx15.2^\circ\) (maybe the given 15.2 is correct).

Step3: Find \(\angle A\) using angle sum

The sum of angles in a triangle is \(180^\circ\). So \(m\angle A = 180^\circ - m\angle B - m\angle C = 180 - 141.8 - 15.2 = 23^\circ\) (matches the given 23).

Step4: Find \(BC\) (side \(a\)) using Law of Sines

\(\frac{BC}{\sin A}=\frac{AC}{\sin B}\). So \(BC=\frac{AC\sin A}{\sin B}\). \(\sin A=\sin(23^\circ)\approx0.3907\), \(AC = 35.2\), \(\sin B\approx0.618\). Then \(BC=\frac{35.2\times0.3907}{0.618}\approx\frac{13.75}{0.618}\approx22.25\approx22.1\) (matches the given 22.1).

Answer:

• \(m\angle C = 15.2^\circ\)
• \(m\angle A = 23^\circ\)
• \(BC = 22.1\) cm