Question

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

Explanation:

Step1: Recall the cosine formula

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Given \(\cos(22.6^{\circ})=\frac{b}{13}\), we can solve for \(b\): \(b = 13\times\cos(22.6^{\circ})\approx13\times\frac{12}{13}=12\) (since \(\cos(22.6^{\circ})\approx\frac{12}{13}\)).

Step2: Recall the tangent formula

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For angle \(A = 22.6^{\circ}\), the opposite side is \(a\) and the adjacent side is \(b\). Since \(b = 12\), the formula \(\tan(22.6^{\circ})=\frac{a}{b}\) becomes \(\tan(22.6^{\circ})=\frac{a}{12}\) or \(\tan(22.6^{\circ})=\frac{a}{b}\) (where \(b = 12\)).

Answer:

\(\tan(22.6^{\circ})=\frac{a}{12}\) (the first option in the left - hand side of the right - hand part of the image, assuming the first option in the left is \(\tan(22.6^{\circ})=\frac{a}{12}\) after substituting \(b = 12\))