Question

question 8 of 10
the measure of \\( \theta \\), in degrees, is approximately \\( 48.8^{\circ} \\).

a. true
b. false

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In this triangle, let \(a = 4\), \(b=7\), \(c = 5.3\). We want to find \(\theta\) (the angle between sides \(a = 4\) and \(b=7\)). So we can rearrange the Law of Cosines to solve for \(\cos\theta\):
\(\cos\theta=\frac{a^{2}+b^{2}-c^{2}}{2ab}\)
Substitute \(a = 4\), \(b = 7\), \(c=5.3\) into the formula:
\(a^{2}=4^{2}=16\), \(b^{2}=7^{2}=49\), \(c^{2}=5.3^{2}=28.09\)
\(\cos\theta=\frac{16 + 49-28.09}{2\times4\times7}\)

Step2: Calculate the numerator and denominator

First, calculate the numerator: \(16 + 49-28.09=36.91\)
Then, calculate the denominator: \(2\times4\times7 = 56\)
So \(\cos\theta=\frac{36.91}{56}\approx0.6591\)

Step3: Find the angle \(\theta\)

To find \(\theta\), we take the inverse cosine (arccos) of \(0.6591\):
\(\theta=\arccos(0.6591)\)
Using a calculator, \(\arccos(0.6591)\approx48.8^{\circ}\)

Answer:

A. True