Question

a right circular cone has a height of 12 meters and a slant length of 13 meters, as shown in the figure above. which of the following equations will provide the solution for the measure of $\theta$, the angle formed by the slant and the cones radius?
a $cos \theta=\frac{12}{13}$
b $cos \theta=\frac{13}{12}$
c $sin \theta=\frac{12}{13}$
d $sin \theta=\frac{13}{12}$
e $\tan \theta=\frac{12}{13}$

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Identify sides relative to \(\theta\)

The height of the cone (\(12\) meters) is the side opposite to \(\theta\), and the slant height (\(13\) meters) is the hypotenuse.

Step3: Apply the sine ratio

Using the formula \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), we get \(\sin\theta=\frac{12}{13}\).

Answer:

C. \(\sin\theta=\frac{12}{13}\)