Question

purple: a right triangle with one acute angle 32°, the adjacent side to 32° is 12, and the opposite side is x.

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle with angle \(32^\circ\), adjacent side \(12\), and opposite side \(x\). The tangent function relates opposite and adjacent: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).
\(\tan(32^\circ)=\frac{x}{12}\)

Step2: Solve for \(x\)

Multiply both sides by \(12\): \(x = 12\times\tan(32^\circ)\)
Calculate \(\tan(32^\circ)\approx0.6249\), so \(x\approx12\times0.6249 = 7.4988\approx7.5\)

Answer:

\(x\approx7.5\)