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