Question

the picture below shows a right - triangle - shaped charging stand for a gaming system: image of right triangle with right angle at c, bc = 3 in, angle at b is 50°, vertices b, c, a which expression shows the height, ac, of the charging stand? options: $\frac{3}{cos 50^circ}$, $3(\tan 50^circ)$, $\frac{3}{\tan 50^circ}$, $3(cos 50^circ)$

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\) (right - angled at \(C\)), we know that for an acute angle \(\theta = 50^{\circ}\), the tangent of the angle is defined as the ratio of the opposite side to the adjacent side. The formula for tangent is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, the angle at \(B\) is \(50^{\circ}\), the adjacent side to \(50^{\circ}\) is \(BC = 3\) in, and the opposite side to \(50^{\circ}\) is \(AC\) (the height we want to find).

So, \(\tan(50^{\circ})=\frac{AC}{BC}\)

Step2: Solve for \(AC\)

We know that \(BC = 3\) in. Substituting \(BC = 3\) into the tangent formula \(\tan(50^{\circ})=\frac{AC}{3}\). To solve for \(AC\), we multiply both sides of the equation by \(3\).

\(AC = 3\times\tan(50^{\circ})\) or \(AC = 3(\tan50^{\circ})\)

Answer:

\(3(\tan50^{\circ})\) (corresponding to the option "3(tan 50°)")