To determine the slant height of a cone using trigonometry, we can apply the Pythagorean theorem, utilizing the radius and height of the cone.

The slant height of a cone is defined as the distance from the apex (the top point) of the cone to any point on the edge of its circular base. To compute this length using trigonometric principles, it is essential to know the radius $r$ of the base and the perpendicular height $h$ of the cone. These three measurements form a right-angled triangle, where the slant height $l$ serves as the hypotenuse.

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For our cone, this relationship can be expressed as:

To find the slant height $l$, we can rearrange the formula to isolate $l$:

For instance, consider a cone with a base radius of $3 \, \text{cm}$ and a height of $4 \, \text{cm}$. We can calculate the slant height as follows:

Thus, the slant height of the cone is $5 \, \text{cm}$. This method effectively utilizes fundamental principles of trigonometry and the Pythagorean theorem to efficiently determine the slant height.