Question

solve the triangle.
a = 14.0 ft, b = 17.0 ft, c = 24.0 ft
what is the measure of angle a?
a = \\(\square\\)°
(round to two decimal places as needed.)
what is the measure of angle b?
b = \\(\square\\)°
(round to two decimal places as needed.)
what is the measure of angle c?
c = \\(\square\\)°
(round to two decimal places as needed.)

Explanation:

Step1: Use Law of Cosines for angle A

The Law of Cosines formula for angle \( A \) is \( \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc} \). Substitute \( a = 14.0 \), \( b = 17.0 \), \( c = 24.0 \):
\( \cos A=\frac{17.0^{2}+24.0^{2}-14.0^{2}}{2\times17.0\times24.0}=\frac{289 + 576-196}{816}=\frac{669}{816}\approx0.82 \)
Then \( A=\arccos(0.82)\approx34.92^{\circ} \)

Step2: Use Law of Cosines for angle B

Law of Cosines for angle \( B \): \( \cos B=\frac{a^{2}+c^{2}-b^{2}}{2ac} \)
Substitute values: \( \cos B=\frac{14.0^{2}+24.0^{2}-17.0^{2}}{2\times14.0\times24.0}=\frac{196 + 576 - 289}{672}=\frac{483}{672}\approx0.72 \)
\( B=\arccos(0.72)\approx43.94^{\circ} \)

Step3: Use angle sum property for angle C

In a triangle, \( A + B + C=180^{\circ} \), so \( C = 180^{\circ}-A - B \)
Substitute \( A\approx34.92^{\circ} \), \( B\approx43.94^{\circ} \): \( C=180 - 34.92 - 43.94 = 101.14^{\circ} \)

Answer:

Angle \( A\approx34.92^{\circ} \)
Angle \( B\approx43.94^{\circ} \)
Angle \( C\approx101.14^{\circ} \)