Question

question 14 (mandatory) (1 point)
in \\( \triangle a b c, a = 5.4 \mathrm { m }, b = 7.2 \mathrm { m } \\), and \\( c = 10.0 \mathrm { m } \\). determine \\( \angle c \\) to the nearest degree.
\\( \bigcirc \\) a) \\( 97 ^ { \circ } \\)
\\( \bigcirc \\) b) \\( 92 ^ { \circ } \\)
\\( \bigcirc \\) c) \\( 104 ^ { \circ } \\)
\\( \bigcirc \\) d) \\( 108 ^ { \circ } \\)

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\).
We can re - arrange it to solve for \(\cos C\): \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}\).
Substitute \(a = 5.4\), \(b = 7.2\), and \(c = 10.0\) into the formula:
\(\cos C=\frac{(5.4)^{2}+(7.2)^{2}-(10.0)^{2}}{2\times5.4\times7.2}\)
First, calculate the numerator: \((5.4)^{2}=29.16\), \((7.2)^{2}=51.84\), \((10.0)^{2}=100\).
\(29.16 + 51.84-100=81 - 100=-19\)
Then, calculate the denominator: \(2\times5.4\times7.2 = 77.76\)
So, \(\cos C=\frac{-19}{77.76}\approx - 0.2443\)

Step2: Find the angle \(C\)

Since \(C=\cos^{-1}(-0.2443)\)
Using a calculator, \(C\approx104^{\circ}\)

Answer:

C. \(104^{\circ}\)