# Solving problems using trigonometry - slope angle

In this class of problems, we are given a slope or ramp with some dimensions known, and we are asked to find the angle of the slope or ramp.

### Problem:

A ramp has been built to make a stage wheelchair accessible. The building inspector needs to find the angle of the ramp to see if it meets regulations. He has no instrument for measuring angles. With a tape measure, he sees the stage is 4t high and the distance along the ramp is 28ft.

## Step 1. Draw a diagram

Include all the information given and label the measure we are asked to find as x. Draw it as close to scale as you can. ## Step 2. Find right triangles

We can assume the side of the stage is vertical and makes a right angle at the floor (point C). So the ramp itself is a right triangle (ABC).

## Step 3. Choose a tool

Right Triangle Toolbox Reviewing what we are given and what we need:

• We are asked to find x, the angle at which the ramp goes up to the stage.
• We are given the hypotenuse (AB) and the side opposite the angle
Looking at our toolbox, we are looking for a function that involves an angle, its opposite side and the hypotenuse. We see that the sin function meets our needs: where O = the side Opposite the angle, H is the Hypotenuse.

## Step 4. Solve the equation

Inserting the values given and the unknown x: Using a calculator, divide 4 by 28: What angle has 0.1429 as its sine? For this we use the inverse function arcSine. It tells us what angle has a given sine.

Using a calculator* again, we find that arcSin(0.1429) is 8.22°, so

x = 8.22°
* Note: On calculators and spreadsheets, arcSin is sometimes called asin or sin-1.

## Step 5. Is it reasonable?

We see from our calculation that the ramp angle is somewhere around 9°. Looking at our diagram we see this looks about right.
If you get a very different answer, the most common error is not setting the calculator to work in degrees or radians as needed.

## Try it yourself

Repeat this problem with a stage height of 8ft. The ramp angle should come out to about 16.6°.

## See it in reverse

See this example where the angle and stage height are known but the ramp length is not.