Question

triangle abc is a right triangle and cos(22.6°)=\frac{b}{13}. solve for b and round to the nearest whole number. which equation correctly uses the value of b to solve for a? o tan(22.6°)=\frac{a}{13} o tan(22.6°)=\frac{13}{a} o tan(22.6°)=\frac{a}{12} o tan(22.6°)=\frac{12}{a}

Explanation:

Step1: Recall cosine definition

In right - triangle \(ABC\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Given \(\cos(22^{\circ})=\frac{b}{13}\), we can solve for \(b\) as \(b = 13\times\cos(22^{\circ})\).
Using a calculator, \(\cos(22^{\circ})\approx0.9272\), so \(b = 13\times0.9272=12.0536\approx12\).

Step2: Recall tangent definition

In right - triangle \(ABC\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). The angle \(\theta = 22^{\circ}\), the opposite side to the \(22^{\circ}\) angle is \(a\) and the adjacent side is \(b\). We know \(b\approx12\) and the hypotenuse is \(13\). By the Pythagorean theorem \(a=\sqrt{13^{2}-b^{2}}\). Also, \(\tan(22^{\circ})=\frac{a}{b}\). Since \(b\approx12\), \(\tan(22^{\circ})=\frac{a}{12}\).

Answer:

\(\tan(22^{\circ})=\frac{a}{12}\)