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

We know $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta=22.6^\circ$, adjacent side = $b$, hypotenuse = 13. Rearrange to solve for $b$:
$b = 13 \times \cos(22.6^\circ)$
Calculate $\cos(22.6^\circ) \approx 0.923$, so $b \approx 13 \times 0.923 = 12.0$, rounded to whole number: $b=12$

Step2: Identify sides for tangent

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

Answer:

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