Question

1. \\(\tan c\\)

\\(\

$$\begin{tikzpicture}scale=0.5 \\coordinate (b) at (0,0); \\coordinate (c) at (0,24); \\coordinate (a) at (32,0); \\draw (b) -- (c) -- (a) -- (b); \\draw (b) nodebelow left {$b$}; \\draw (c) nodeabove left {$c$}; \\draw (a) nodebelow right {$a$}; \\draw (b) -- (c) nodeleft {24}; \\draw (b) -- (a) nodebelow {32}; \\draw (c) -- (a) nodeabove {40}; \\draw (b) nodebelow left {$\\angle$}; \\filldraw (b) circle (2pt); \\filldraw (c) circle (2pt); \\filldraw (a) circle (2pt); \\end{tikzpicture}$$

\\)

Explanation:

Step1: Recall tangent definition

For right triangle, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Step2: Identify sides for $\angle C$

Opposite to $\angle C$ is $AB=32$, adjacent is $BC=24$

Step3: Substitute values and simplify

$\tan C = \frac{32}{24} = \frac{4}{3}$

Answer:

$\frac{4}{3}$