Question

solve the triangle.
a = 4.43 m c = 5.78 m b = 26.1°
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?
47.3°
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)
what is the measure of angle c?
106.6°
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)

Explanation:

Step1: Use the Law of Cosines to find side b

The Law of Cosines states that \( b^2 = a^2 + c^2 - 2ac \cos B \). We know \( a = 4.43 \) m, \( c = 5.78 \) m, and \( B = 26.1^\circ \). First, calculate \( \cos(26.1^\circ) \approx 0.8975 \). Then:
\( a^2 = 4.43^2 = 19.6249 \)
\( c^2 = 5.78^2 = 33.4084 \)
\( 2ac = 2 \times 4.43 \times 5.78 = 2 \times 25.6054 = 51.2108 \)
\( 2ac \cos B = 51.2108 \times 0.8975 \approx 46.0617 \)
Now, \( b^2 = 19.6249 + 33.4084 - 46.0617 = 53.0333 - 46.0617 = 6.9716 \)
Take the square root: \( b = \sqrt{6.9716} \approx 2.6404 \)

Step2: Verify angle A (optional, but given)

Using the Law of Sines: \( \frac{\sin A}{a} = \frac{\sin B}{b} \). We know \( a = 4.43 \), \( b \approx 2.6404 \), \( B = 26.1^\circ \). \( \sin(26.1^\circ) \approx 0.4409 \). Then \( \sin A = \frac{4.43 \times 0.4409}{2.6404} \approx \frac{1.9532}{2.6404} \approx 0.740 \), so \( A \approx \arcsin(0.740) \approx 47.3^\circ \), which matches the given value.

Step3: Verify angle C (optional, but given)

Since the sum of angles in a triangle is \( 180^\circ \), \( A + B + C = 180^\circ \). So \( C = 180 - 47.3 - 26.1 = 106.6^\circ \), which matches the given value.

Answer:

The length of side b is \(\boxed{2.6404}\) m.