Question

solve the triangle.
a = 10.0 ft, b = 13.0 ft, c = 19.0 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 \( a = 10.0 \), \( b = 13.0 \), \( c = 19.0 \):
\( \cos A=\frac{13.0^{2}+19.0^{2}-10.0^{2}}{2\times13.0\times19.0}=\frac{169 + 361- 100}{494}=\frac{430}{494}\approx0.8704 \)
Then \( A=\arccos(0.8704)\approx29.54^{\circ} \)

Step2: Use Law of Cosines for angle B

The formula for angle \( B \) is \( \cos B=\frac{a^{2}+c^{2}-b^{2}}{2ac} \). Substitute values:
\( \cos B=\frac{10.0^{2}+19.0^{2}-13.0^{2}}{2\times10.0\times19.0}=\frac{100 + 361 - 169}{380}=\frac{292}{380}\approx0.7684 \)
Then \( B=\arccos(0.7684)\approx39.79^{\circ} \)

Step3: Use angle sum property for angle C

The sum of angles in a triangle is \( 180^{\circ} \). So \( C = 180^{\circ}-A - B \).
Substitute \( A\approx29.54^{\circ} \), \( B\approx39.79^{\circ} \):
\( C\approx180 - 29.54 - 39.79 = 110.67^{\circ} \)

Answer:

Angle \( A \approx \boldsymbol{29.54^{\circ}} \)
Angle \( B \approx \boldsymbol{39.79^{\circ}} \)
Angle \( C \approx \boldsymbol{110.67^{\circ}} \)