**Finding Length in Trigonometry **

**I**n this lesson, we use the basic trigonometric functions to solve word problems. In these problems, we use an angle and a distance to find the other distances around a right triangle.

It is fortunate for us that right triangles appear in many situations. Walls, poles, and buildings are built at right angles to the ground. Trees grow upward at right angles to the ground (usually). Nearly everything that human beings make involves straight lines and right angles. Even where no lines exist, such as a plane flying in the air, we can imagine a line running straight down to the ground, giving the altitude and making a right angle with the ground.

The trick to solving such problems is to sketch the right triangle. Usually, the ground forms one of the sides. The right angle is then formed by the wall, height, pole, tree, or something else that rises straight up. The length that is given and the one that is sought will be either *O* and *H*, *A* and *H*, or *O* and *A*. The angle *θ* that is given will thus be sin(*θ*) = , cos(*θ*) = , or tan(*θ*) = . Write out this equation, then solve for the desired length, using a calculator to estimate the trigonometric function.

**Example 1**

Suppose a 20-foot ladder leans against a wall. If it makes a 75° angle with the ground, how high up does it reach on the wall?

We suppose that the wall rises up at a right angle to the ground, and so we sketch Figure 13.1.

The side that is given is the hypotenuse *H* = 20 feet, and the side that we want to know is the opposite *O*, which we label *x*. Thus, we use the sine function.

sin(*θ*) =

sin(75°) =

*x* = 20 sin(75°) ≈ 19.3 feet

Thus, the ladder will rest at a spot 19.3 feet up the wall. Note: Because there are no absolute rules about where to round off, the answer could be 19 feet, 19.32 feet, 19.319 feet, or anything similar.

**Example 2**

The manufacturer of a ladder decides that the ladder should never be used at more than an 80° angle to the ground. When a 35-foot ladder is leaned against a wall, how close can the bottom of the ladder be from the wall?

Here, we sketch the triangle in Figure 13.2. We want to know the length of adjacent side *A* when the hypotenuse is *H* = 35 feet.

cos(*θ*) =

cos(80°) =

*x* = 35 cos(80°) ≈ 6.08 feet

The manufacturer should insist that people keep the base of the ladder 6.08 feet (or, just more than 6 feet) from the base of the wall to avoid accidents.

**Example 3**

Suppose a painter wants a ladder to rest against a spot 40 feet up a wall. If the ladder will make a 72° angle with the ground, how long must the ladder be?

We sketch the triangle in Figure 13.3. Here, the side opposite the angle is *O* = 40 feet long and the hypotenuse *H* is the unknown length.

sin(θ) =

sin(72°) =

*x* = ≈ 42 feet

The ladder must be 42 feet long.

** Example 4**

When a person stands 100 feet away from a skyscraper, the top of the building appears to be at an 82° angle from horizontal. How tall is the building?

We sketch the triangle in Figure 13.4. Here, we know that the length of the adjacent side is *A* = 100 feet, but we want to know the length of the opposite side *O*.

tan(*θ*) =

tan(82°) =

*x* = 100 tan(82°) ≈ 712 feet

The building is approximately 712 feet tall.

**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,

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.

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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