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 formula for side \( b \) is \( b^2 = a^2 + c^2 - 2ac \cos B \).
Given \( a = 4.43 \, \text{m} \), \( c = 5.78 \, \text{m} \), and \( B = 26.1^\circ \).
First, calculate \( \cos(26.1^\circ) \approx 0.8975 \).
Then, \( a^2 = 4.43^2 = 19.6249 \), \( c^2 = 5.78^2 = 33.4084 \), and \( 2ac = 2 \times 4.43 \times 5.78 = 50.7348 \).
So, \( b^2 = 19.6249 + 33.4084 - 50.7348 \times 0.8975 \).
Calculate \( 50.7348 \times 0.8975 \approx 45.534 \).
Then, \( b^2 = 19.6249 + 33.4084 - 45.534 = 7.4993 \).
Take the square root: \( b = \sqrt{7.4993} \approx 2.7385 \).

Answer:

The length of side \( b \) is \( 2.7385 \) m.