Finding Length in Trigonometry Study Guide (page 2)

Updated on Oct 2, 2011

Example 5

A window is 45 feet off the ground. A person looks out and sees a car on the road. If the car appears to be 10° below horizontal, how far away is the car from the building?

We sketch Figure 13.5. There are two ways to look at this situation. We could use the top triangle, with a 10° angle, an opposite side of O = 45 feet, and an unknown adjacent side A. In this case,

Figure 13.5

tan(θ) =
tan(10°) =
x = ≈ 255 feet

If we use the lower triangle, then the angle at the window has to be 80° because the height of the wall makes a right angle (90°) with the horizontal. If the diagonal is 10° down from horizontal, it must also be 80° up from the wall. Now, the unknown side is the opposite side O, and the known side is the adjacent A = 45 feet.

tan(θ) =
tan(80°) =
x = 45 tan(80°) ≈ 255 feet

Both methods reach the same conclusion: that the car is about 255 feet away from the building.

In the last problem, we wanted to know the distance from the car to the building along the ground. If we had wanted to know the direct distance through the air from the person in the window to the car, then we would have used the hypotenuse H as the unknown x. Usually, distances are measured along the ground.

Example 6

A 5-foot-tall photographer wants to view a 20-foot-tall tree so that the top of the tree appears within 30° of horizontal. How far away from the tree must the photographer stand?

We sketch Figure 13.6. Because the photographer is 5 feet tall, we imagine that the camera, aimed horizontally, points at a spot on the tree about 5 feet off the ground. Only the top 15 feet of the tree must be fit into the 30° angle. Thus, the opposite side has length O = 15 feet and the adjacent side A is what we want to know.

Figure 13.6

tan(θ) =
tan(30°) =
x = ≈ 26 feet

The photographer should stand back about 26 feet.

Practice problems for this study guide can be found at:

Finding Length in Trigonometry Practice Questions

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