Question

solve the triangle.
a = 4.29 m c = 5.94 m b = 25.1°

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?
41.5°
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)
what is the measure of angle c?
113.4°
(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 for a triangle with sides \(a\), \(b\), \(c\) and the angle \(B\) opposite side \(b\), \(b^{2}=a^{2}+c^{2}-2ac\cos(B)\).
Given \(a = 4.29\) m, \(c = 5.94\) m, and \(B=25.1^{\circ}\).
First, calculate \(a^{2}\), \(c^{2}\), and \(2ac\cos(B)\):
\(a^{2}=(4.29)^{2}=18.4041\)
\(c^{2}=(5.94)^{2}=35.2836\)
\(2ac = 2\times4.29\times5.94=2\times25.4826 = 50.9652\)
\(\cos(25.1^{\circ})\approx\cos(25.1)\approx0.9056\)
Then \(2ac\cos(B)=50.9652\times0.9056\approx46.1541\)
Now, \(b^{2}=a^{2}+c^{2}-2ac\cos(B)=18.4041 + 35.2836-46.1541=7.5336\)
Take the square root of \(b^{2}\) to find \(b\): \(b=\sqrt{7.5336}\approx2.7447\)

Step2: Verify angle A (optional, but to check)

We can use the Law of Sines: \(\frac{\sin(A)}{a}=\frac{\sin(B)}{b}\)
\(\sin(A)=\frac{a\sin(B)}{b}=\frac{4.29\times\sin(25.1^{\circ})}{2.7447}\)
\(\sin(25.1^{\circ})\approx0.4245\)
\(4.29\times0.4245\approx1.8211\)
\(\sin(A)=\frac{1.8211}{2.7447}\approx0.6635\)
\(A=\arcsin(0.6635)\approx41.5^{\circ}\) (matches the given value)

Step3: Verify angle C (optional, but to check)

Since the sum of angles in a triangle is \(180^{\circ}\), \(C = 180^{\circ}-A - B=180 - 41.5-25.1 = 113.4^{\circ}\) (matches the given value)

Answer:

The length of side \(b\) is \(\boxed{2.7447}\) m.