A 21-foot ladder is leaning against a building. The base of the ladder is 7 feet from the base of the building. To the nearest degree, what is the measure of the angle that the ladder makes with the ground? (Hint: Sketch the problem before solving.)
Have you sketched the problem?
Yes.
In that case, review your basic trig functions.
cos(x) = 7/21
Now just find x, with that value for cosine.
To solve this problem, we need to use trigonometry, specifically the sine function. Let's start by drawing a sketch of the problem.
Imagine a right triangle formed by the ladder, the ground, and the side of the building. The ladder represents the hypotenuse of the triangle, and the distance from the base of the ladder to the building represents one of the legs. We are looking for the angle between the hypotenuse (the ladder) and the ground.
Let's label the sides of the triangle:
- The length of the ladder (hypotenuse) is 21 feet.
- The distance from the base of the ladder to the building (leg) is 7 feet.
Now, let's use the sine function to find the angle. The sine of an angle can be calculated by dividing the opposite side by the hypotenuse.
In this case, the opposite side is the distance from the base of the ladder to the building, which is 7 feet. The hypotenuse is the length of the ladder, which is 21 feet.
So, the sine of the angle (θ) can be written as:
sin(θ) = opposite / hypotenuse
= 7 / 21
Now, we can calculate the value of sin(θ):
sin(θ) ≈ 0.33
To find the angle θ, we need to take the inverse sine (also called arcsine or sin^-1) of 0.33:
θ ≈ sin^-1(0.33)
Using a calculator in degree mode, we find:
θ ≈ 19.47 degrees
Therefore, to the nearest degree, the measure of the angle that the ladder makes with the ground is approximately 19 degrees.