A ladder 10 ft long rests against a vertical wall. Let theta be the angle between the top of the ladder and the wall and let x be the distance.

In this scenario, we have a classic right triangle formed by the ladder, the wall, and the ground. The ladder itself serves as the hypotenuse, which is 10 feet long. The height at which the ladder touches the wall can be represented as the opposite side of the triangle, and the horizontal distance from the base of the ladder to the wall is represented as the adjacent side.

We can use trigonometric functions to relate the angle theta (θ) to these sides. Specifically, the sine of the angle theta is equal to the opposite side (the height) divided by the hypotenuse (the length of the ladder). Therefore, we can write:

sin(θ) = opposite / hypotenuse

Substituting the known length of the ladder gives us:

sin(θ) = height / 10

Similarly, the cosine of theta relates the adjacent side (the distance x) to the hypotenuse:

cos(θ) = adjacent / hypotenuse

This can be expressed as:

cos(θ) = x / 10

Using the Pythagorean theorem, we also have:

(height)² + (x)² = (10)²

This equation provides a complete relationship between the three sides of the triangle and allows us to solve for the height or the distance x depending on the angle θ. By understanding these relationships, you can solve for various unknowns as you alter the angle of the ladder against the wall.