Question

solve the triangle.
a = 7.498 in c = 6.930 in b = 76.57°

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 formula is \( b^{2}=a^{2}+c^{2}-2ac\cos B \).
Substitute \( a = 7.498 \), \( c = 6.930 \), and \( B=76.57^{\circ} \)

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

The Law of Sines formula is \( \frac{\sin A}{a}=\frac{\sin B}{b} \)
\( \sin A=\frac{a\sin B}{b} \)
Substitute \( a = 7.498 \), \( b\approx8.963 \), and \( B = 76.57^{\circ} \)

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Step3: Find angle \( C \) using the angle - sum property of a triangle

Since \( A + B+C=180^{\circ} \), then \( C=180^{\circ}-A - B \)
Substitute \( A\approx54.59^{\circ} \) and \( B = 76.57^{\circ} \)

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Answer:

The length of side \( b \) is \( 8.963 \) in.
The measure of angle \( A \) is \( 54.59^{\circ} \).
The measure of angle \( C \) is \( 48.84^{\circ} \).