Question

triangle abc is a right triangle and \\(\cos(22.6^\circ)=\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\\)? \\(\bigcirc\\) \\(\tan(22.6^\circ) = \frac{a}{13}\\) \\(\bigcirc\\) \\(\tan(22.6^\circ) = \frac{13}{a}\\) \\(\bigcirc\\) \\(\tan(22.6^\circ) = \frac{a}{12}\\) \\(\bigcirc\\) \\(\tan(22.6^\circ) = \frac{12}{a}\\)

Explanation:

Step1: Solve for $b$ using cosine formula

Rearrange $\cos(22.6^\circ)=\frac{b}{13}$ to isolate $b$:
$b = 13\times\cos(22.6^\circ)$
Calculate $\cos(22.6^\circ)\approx0.923$, so $b\approx13\times0.923=12.0$

Step2: Identify sides for tangent formula

For $\angle A=22.6^\circ$, $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}$. Substitute $b=12$:
$\tan(22.6^\circ)=\frac{a}{12}$

Answer:

First part (value of $b$): 12
Second part (correct equation): $\boldsymbol{\tan(22.6^\circ) = \frac{a}{12}}$