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# Solving problems using trigonometry - slant distance

In this class of problems, we are given an angle and some other measures, and asked to find the distance up a slope or ramp.

### Problem:

We are designing a ramp up to a stage to make it wheelchair accessible. The stage is 4 ft high, and building regulations state that the ramp angle must be 9°. How long will the ramp be, measured along its slope?

## 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 hypotenuse of the right triangle ABC.
• We are given an angle and the side opposite that angle
Looking at our toolbox we see that the sin function uses all three of these: 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 we see that sin(9°) is 0.1564, so Transposing: (multiply both side by x, divide both sides by 0.1564) Dividing:

## Step 5. Is it reasonable?

We see from our calculation that the ramp length is roughly 6 times the stage height. 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 length should come out to double the above.

Why?    All 9° right triangles are similar (AAA), and in similar triangles all corresponding sides remain in the same ratio. So if you double one side (the side BC) the others will all double also.

## See it in reverse

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