Question

determine the missing angle measures using the law of cosines, law of sines, and other properties of triangles. note: ∠b is an obtuse angle. please make sure to think about this when determining your answer. you might want to determine m∠a and m∠c first.
a. m∠a = 40.4 degrees
b. m∠b = enter a mathematical expression more.. degrees
c. m∠c = 29.3 degrees
triangle abc is shown below with all three side lengths known.
(image of triangle abc with side ab = 11.3 cm, side ac = 21.7 cm, side bc = 15 cm)
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.
question 8. points possible: 3
unlimited attempts.
score on last attempt: 2 out of 3
score in gradebook: 2 out of 3

Explanation:

Step1: Identify sides and Law of Cosines

In triangle \(ABC\), sides: \(AB = 11.3\) cm, \(BC = 15\) cm, \(AC = 21.7\) cm. Let's label: \(a = BC = 15\), \(b = AC = 21.7\), \(c = AB = 11.3\). Angles: \(\angle A\) opposite \(a\), \(\angle B\) opposite \(b\), \(\angle C\) opposite \(c\).

First, find \(\angle A\) using Law of Cosines: \(\cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\)
Substitute values: \(b = 21.7\), \(c = 11.3\), \(a = 15\)
\(\cos A=\frac{21.7^{2}+11.3^{2}-15^{2}}{2\times21.7\times11.3}\)
Calculate numerator: \(21.7^{2}=470.89\), \(11.3^{2}=127.69\), \(15^{2}=225\)
Numerator: \(470.89 + 127.69 - 225 = 373.58\)
Denominator: \(2\times21.7\times11.3 = 494.62\)
\(\cos A=\frac{373.58}{494.62}\approx0.7553\)
\(\angle A=\arccos(0.7553)\approx40.4^\circ\) (matches given \(m\angle A = 40.4\) degrees)

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

Law of Sines: \(\frac{\sin C}{c}=\frac{\sin A}{a}\)
\(\sin C=\frac{c\sin A}{a}\)
Substitute \(c = 11.3\), \(\sin A=\sin(40.4^\circ)\approx0.648\), \(a = 15\)
\(\sin C=\frac{11.3\times0.648}{15}\approx\frac{7.3224}{15}\approx0.4882\)
\(\angle C=\arcsin(0.4882)\approx29.3^\circ\) (matches given \(m\angle C = 29.3\) degrees)

Step3: Find \(\angle B\) using triangle angle sum

Triangle angle sum: \(m\angle A + m\angle B + m\angle C = 180^\circ\)
\(m\angle B = 180 - 40.4 - 29.3 = 110.3^\circ\) (since \(\angle B\) is obtuse, this makes sense)

Answer:

a. \(m\angle A = 40.4\) degrees
b. \(m\angle B = 110.3\) degrees
c. \(m\angle C = 29.3\) degrees