Question

solve the triangle.
a=4.45 m c=5.78 m b=28.7°
what is the length of side b?
□ m
(simplify your answer. type an integer or a decimal. round to four decimal places if needed.)
what is the measure of angle a?
□°
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)
what is the measure of angle c?
□°
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)

Explanation:

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

The Law of Cosines states that \( b^2 = a^2 + c^2 - 2ac \cos B \).
Given \( a = 4.45 \, \text{m} \), \( c = 5.78 \, \text{m} \), and \( B = 28.7^\circ \).
First, calculate \( \cos(28.7^\circ) \approx 0.8771 \).
Then, \( a^2 = 4.45^2 = 19.8025 \), \( c^2 = 5.78^2 = 33.4084 \), and \( 2ac = 2 \times 4.45 \times 5.78 = 51.022 \).
So, \( b^2 = 19.8025 + 33.4084 - 2 \times 4.45 \times 5.78 \times 0.8771 \)
\( b^2 = 53.2109 - 51.022 \times 0.8771 \)
\( 51.022 \times 0.8771 \approx 44.751 \)
\( b^2 = 53.2109 - 44.751 = 8.4599 \)
\( b = \sqrt{8.4599} \approx 2.9086 \, \text{m} \)

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

The Law of Sines states that \( \frac{\sin A}{a} = \frac{\sin B}{b} \).
So, \( \sin A = \frac{a \sin B}{b} \)
\( \sin B = \sin(28.7^\circ) \approx 0.4808 \)
\( \sin A = \frac{4.45 \times 0.4808}{2.9086} \)
\( 4.45 \times 0.4808 \approx 2.1396 \)
\( \sin A \approx \frac{2.1396}{2.9086} \approx 0.7356 \)
\( A = \arcsin(0.7356) \approx 47.3^\circ \)

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

The sum of angles in a triangle is \( 180^\circ \), so \( C = 180^\circ - A - B \)
\( C = 180^\circ - 47.3^\circ - 28.7^\circ = 104.0^\circ \)

Answer:

Length of side \( b \): \( 2.9086 \, \text{m} \)
Measure of angle \( A \): \( 47.3^\circ \)
Measure of angle \( C \): \( 104.0^\circ \)