Question

solve the triangle.
a = 7.748 in c = 6.072 in b = 70.29°
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 \( a = 7.748 \), \( c = 6.072 \), \( B = 70.29^\circ \).
First, calculate \( \cos(70.29^\circ) \approx 0.337 \).
Then, \( a^2 = 7.748^2 \approx 59.93 \), \( c^2 = 6.072^2 \approx 36.87 \), \( 2ac = 2\times7.748\times6.072 \approx 93.74 \).
So \( b^2 = 59.93 + 36.87 - 93.74\times0.337 \approx 59.93 + 36.87 - 31.59 \approx 65.21 \).
Thus \( b \approx \sqrt{65.21} \approx 8.075 \).

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

The Law of Sines is \( \frac{\sin A}{a} = \frac{\sin B}{b} \).
So \( \sin A = \frac{a \sin B}{b} \).
\( \sin B = \sin(70.29^\circ) \approx 0.941 \).
\( \sin A = \frac{7.748\times0.941}{8.075} \approx \frac{7.29}{8.075} \approx 0.903 \).
Then \( A \approx \arcsin(0.903) \approx 64.62^\circ \).

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

In a triangle, \( A + B + C = 180^\circ \).
So \( C = 180^\circ - A - B \approx 180 - 64.62 - 70.29 = 45.09^\circ \).

Answer:

Length of side \( b \): \( \boxed{8.075} \) in
Measure of angle \( A \): \( \boxed{64.62}^\circ \)
Measure of angle \( C \): \( \boxed{45.09}^\circ \)