Question

which equation can be solved to find one of the missing side lengths in the triangle?

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle, for an angle \(\theta\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, the hypotenuse \(AB = 12\) units, angle at \(B\) is \(60^{\circ}\), side \(a\) (adjacent to \(60^{\circ}\)) and side \(b\) (opposite to \(60^{\circ}\)).

Step2: Apply \(\cos\theta\) ratio for side \(a\)

Using \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), with \(\theta = 60^{\circ}\), adjacent side \(=a\) and hypotenuse \(=12\). So, \(\cos60^{\circ}=\frac{a}{12}\).

Step3: Apply \(\sin\theta\) ratio for side \(b\)

Using \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), with \(\theta = 60^{\circ}\), opposite side \(=b\) and hypotenuse \(=12\). So, \(\sin60^{\circ}=\frac{b}{12}\).

Answer:

Equations \(\cos60^{\circ}=\frac{a}{12}\) or \(\sin60^{\circ}=\frac{b}{12}\) can be used to find the missing side lengths.