Question

solve the triangle.
a = 4.26 m c = 5.66 m b = 25.8°
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 \( a = 4.26 \, \text{m} \), \( c = 5.66 \, \text{m} \), and \( B = 25.8^\circ \).
First, calculate \( \cos(25.8^\circ) \approx 0.9009 \).
Then, \( a^2 = 4.26^2 = 18.1476 \), \( c^2 = 5.66^2 = 32.0356 \), and \( 2ac = 2 \times 4.26 \times 5.66 = 48.3672 \).
So, \( b^2 = 18.1476 + 32.0356 - 48.3672 \times 0.9009 \)
\( b^2 = 50.1832 - 43.583 \approx 6.6002 \)
Then, \( b = \sqrt{6.6002} \approx 2.5691 \, \text{m} \).

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

The Law of Sines states that \( \frac{\sin A}{a} = \frac{\sin B}{b} \).
So, \( \sin A = \frac{a \sin B}{b} \).
We know \( a = 4.26 \), \( \sin B = \sin(25.8^\circ) \approx 0.4359 \), and \( b \approx 2.5691 \).
\( \sin A = \frac{4.26 \times 0.4359}{2.5691} \approx \frac{1.857}{2.5691} \approx 0.7228 \)
Then, \( A = \arcsin(0.7228) \approx 46.3^\circ \).

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

The sum of angles in a triangle is \( 180^\circ \).
So, \( C = 180^\circ - A - B \).
\( C = 180 - 46.3 - 25.8 = 107.9^\circ \).

Answer:

Length of side \( b \): \( \boldsymbol{2.5691} \, \text{m} \)
Measure of angle \( A \): \( \boldsymbol{46.3^\circ} \)
Measure of angle \( C \): \( \boldsymbol{107.9^\circ} \)