Question

question: which of the following equations could be used to find the value of x? the figure is not drawn to scale answer: $\frac{sin(60)}{56}=\frac{sin(x)}{65}$, $x^{2}=65^{2}+60^{2}-2(65)(cos(56))$, $x^{2}=65^{2}+56^{2}-2(65)(cos(60))$, $cos(x)=\frac{56^{2}+60^{2}-65^{2}}{2(56)(60)}$

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In the given triangle, if \(a = 56\), \(b = 65\), \(\theta=60^{\circ}\) and the side opposite to the \(60^{\circ}\) - angle is not \(x\), and the side opposite to angle \(x\) has length \(x\), then \(x^{2}=56^{2}+65^{2}-2(56)(65)\cos(60)\).

Step2: Analyze the Law - of - Sines

The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). If we consider the angles and sides of the given triangle, \(\frac{\sin x}{65}=\frac{\sin60}{56}\) is also a valid relationship.

The correct equations for finding \(x\) are \(\frac{\sin(x)}{65}=\frac{\sin(60)}{56}\) and \(x^{2}=56^{2}+65^{2}-2(56)(65)\cos(60)\).

Answer:

The first option \(\frac{\sin(x)}{65}=\frac{\sin(60)}{56}\) and the fourth - option \(x^{2}=56^{2}+65^{2}-2(56)(65)\cos(60)\) are the equations that could be used to find the value of \(x\).