Question

solve the triangle.
a = 7.601 in c = 6.298 in b = 76.06°
what is the length of side b?
□ in
(round to the nearest thousandth as needed.)
what is the measure of angle a?
□°
(round to the nearest hundredth as needed.)
what is the measure of angle c?
□°
(round to the nearest hundredth as needed.)

Explanation:

Step1: Find side \( b \) using the Law of Cosines

The Law of Cosines states \( b^2 = a^2 + c^2 - 2ac \cos B \).
Given \( a = 7.601 \), \( c = 6.298 \), \( B = 76.06^\circ \).
First, calculate \( \cos(76.06^\circ) \approx 0.2419 \).
Then, \( a^2 = 7.601^2 \approx 57.775 \), \( c^2 = 6.298^2 \approx 39.665 \), \( 2ac = 2 \times 7.601 \times 6.298 \approx 95.997 \).
So, \( b^2 = 57.775 + 39.665 - 95.997 \times 0.2419 \approx 57.775 + 39.665 - 23.22 \approx 74.22 \).
Thus, \( b \approx \sqrt{74.22} \approx 8.615 \).

Step2: Find angle \( A \) using the Law of Sines

The Law of Sines is \( \frac{\sin A}{a} = \frac{\sin B}{b} \).
So, \( \sin A = \frac{a \sin B}{b} \).
\( \sin(76.06^\circ) \approx 0.9703 \), \( a = 7.601 \), \( b \approx 8.615 \).
\( \sin A \approx \frac{7.601 \times 0.9703}{8.615} \approx \frac{7.375}{8.615} \approx 0.856 \).
Thus, \( A \approx \arcsin(0.856) \approx 58.87^\circ \).

Step3: Find angle \( C \) using the triangle angle sum

The sum of angles in a triangle is \( 180^\circ \).
So, \( C = 180^\circ - A - B \approx 180 - 58.87 - 76.06 = 45.07^\circ \).

Answer:

Length of side \( b \): \( \boldsymbol{8.615} \) in
Measure of angle \( A \): \( \boldsymbol{58.87^\circ} \)
Measure of angle \( C \): \( \boldsymbol{45.07^\circ} \)