Question

which of the following equations could be used to find the value of x?
64, 31, 59, x°
\the figure is not drawn to scale\
answer
\\(\cos x = \frac{59^2 + 64^2 - 31^2}{2(59)(64)}\\) \\(\cos x = \frac{64^2 + 31^2 - 59^2}{2(64)(31)}\\)\\(\cos x = \frac{31^2 + 59^2 - 64^2}{2(31)(59)}\\) none of the choices

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(x\) opposite to side \(c\) is given by \(c^{2}=a^{2}+b^{2}-2ab\cos x\), which can be rearranged to \(\cos x=\frac{a^{2}+b^{2}-c^{2}}{2ab}\).

Step2: Identify the sides and angle

In the given triangle, the angle \(x\) is between the sides of length \(64\) and \(31\), and the side opposite to angle \(x\) is \(59\). So, \(a = 64\), \(b = 31\), and \(c = 59\).

Step3: Apply the Law of Cosines formula

Substitute \(a = 64\), \(b = 31\), and \(c = 59\) into the formula \(\cos x=\frac{a^{2}+b^{2}-c^{2}}{2ab}\). We get \(\cos x=\frac{64^{2}+31^{2}-59^{2}}{2(64)(31)}\).

Answer:

\(\cos x=\frac{64^{2}+31^{2}-59^{2}}{2(64)(31)}\) (the second option among the given choices)