Question

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

cos(60°) = \frac{12}{a}

cos(60°) = \frac{12}{b}

cos(60°) = \frac{b}{a}

mark this and return save and exit next

Explanation:

Step1: Recall cosine - ratio in right - triangle

In a right - triangle, the cosine of an acute angle is defined as the ratio of the adjacent side to the hypotenuse. In right - triangle \(ABC\) with right - angle at \(C\), for angle \(B = 60^{\circ}\), the adjacent side to angle \(B\) is side \(a\) and the hypotenuse is side \(AB = 12\) units.
The formula for cosine of an angle \(\theta\) in a right - triangle is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Apply the cosine formula

For \(\theta = 60^{\circ}\), \(\cos(60^{\circ})=\frac{a}{12}\), which can be rewritten as \(a = 12\cos(60^{\circ})\). If we want to solve for \(a\) from the given options, we can cross - multiply the correct equation. The correct equation based on the cosine ratio is \(\cos(60^{\circ})=\frac{a}{12}\), which is equivalent to \(\cos(60^{\circ})=\frac{12}{ \frac{12}{a}}\). The correct form of the cosine equation for this triangle is \(\cos(60^{\circ})=\frac{a}{12}\), and if we rewrite the options in a standard form, we know that \(\cos(60^{\circ})=\frac{\text{adjacent to }60^{\circ}}{\text{hypotenuse}}\). The adjacent side to the \(60^{\circ}\) angle is \(a\) and the hypotenuse is \(12\). So the correct equation is \(\cos(60^{\circ})=\frac{a}{12}\), or rewritten as \(a = 12\cos(60^{\circ})\). Among the given options, the correct one is \(\cos(60^{\circ})=\frac{a}{12}\) (equivalent to the first option when cross - multiplied correctly).

Answer:

\(\cos(60^{\circ})=\frac{a}{12}\) (equivalent to the first option \(\cos(60^{\circ})=\frac{12}{ \frac{12}{a}}\) after proper cross - multiplication)