Ramp-Up Exercises: Finding the Height of a Lighthouse

To show how the Ramp-Up tool works in practice, let’s walk through a simple example.

I started with this target problem:
"The top of a lighthouse can be seen at an angle of 32 degrees from a boat 100 metres from the base of the tower. How tall is the lighthouse?"

I marked this as "Basic knowledge, needs refreshment" and selected "short-term preparation for exam" from the dropdown. Based on those inputs, Ramp-Up generated a personalized sequence of exercises, like steps on a staircase, designed to guide the student from their current understanding to confidently solving this problem.

Here are the results:

Exercise 1: Recalling Basic Trigonometric Ratios

Match each trigonometric ratio with its correct formula:

1. Sine (sin) = ?

2. Cosine (cos) = ?

3. Tangent (tan) = ?

Options:

• Opposite ÷ Adjacent

• Opposite ÷ Hypotenuse

• Adjacent ÷ Hypotenuse

Remember: SOH-CAH-TOA helps you recall these ratios!

• Sin = Opposite ÷ Hypotenuse

• Cos = Adjacent ÷ Hypotenuse

• Tan = Opposite ÷ Adjacent

Exercise 2: Identifying Triangle Parts

Editor’s Note: Image creation tools are on the way! In the meantime, we recommend quickly sketching an image by hand or using the Visuals Finder tool to illustrate your point.

                C
                |
                |  
                |  height (BC)
                |
                |
angle θ         |
    A ---------- B
        distance (AB)

In the triangle above, if angle θ is at point A:

1. Which side is opposite to angle θ?

2. Which side is adjacent to angle θ?

3. Which trigonometric ratio would you use to find the length of side BC if you know angle θ and the length of AB?

Exercise 3: Calculator Practice

Use your calculator to find the following values (make sure your calculator is in degree mode):

1. tan(30°) = ?

2. tan(45°) = ?

3. tan(60°) = ?

Exercise 4: Simple Application

A ladder leans against a wall, making an angle of 40° with the ground. If the ladder's foot is 3 meters from the wall, how high up the wall does the ladder reach?

1. Draw a right triangle representing this situation

2. Label the known angle and side

3. Identify which trigonometric ratio to use

4. Calculate the height

Exercise 5: Angle of Elevation

From a point on the ground, the angle of elevation to the top of a flagpole is 25° if you are standing 20 meters from the base of the flagpole. How tall is it?

Hint: The angle of elevation is the angle between the horizontal line of sight and the line of sight to the object above.

                   Top
                    |
                    |
                    | height
                    |
                    |
                    |
 Observer --------- Base
                    distance
angle = 25°
distance = 20 m

Exercise 6: Finding Height with Tangent

A 1.75m tall person, whose eyes are at 1.6 meters height, stands 50 meters away from a building and looks up at the top with an angle of elevation of 35°. How tall is the building?

Quick Check:

• What forms the opposite side? (The height)

• What forms the adjacent side? (The distance from the person to the building)

• Which ratio relates these sides? (Tangent)

                   Top
                    |
                    |
                    | height
                    | (to find)
                    |
                    |
 Person ---------- Base
                   distance
angle = 35°
distance = 50 m

Exercise 7: Lighthouse Problem with Guided Steps

The top of a lighthouse can be seen at an angle of 32 degrees from a boat 100 metres from the base of the tower. How tall is the lighthouse?

Step 1: Draw a right triangle representing the situation.

                   Top
                    |
                    |
                    | height
                    | (to find)
                    |
                    |
  Boat ---------- Base
                  distance
angle = 32°
distance = 100 m

Step 2: Label what you know:

• The angle of elevation is 32°

• The distance from the boat to the base is 100 m

Step 3: Identify which trigonometric ratio to use. Since you know the angle and the adjacent side (distance) and need to find the opposite side (height), you should use:

tan(θ) = opposite / adjacent

Step 4: Substitute the values and solve:

tan(32°) = height / 100

height = 100 × tan(32°)

Step 5: Calculate the final answer using your calculator (in degree mode).

Exercise 8: Target Problem/Original Problem

The top of a lighthouse can be seen at an angle of 32 degrees from a boat 100 metres from the base of the tower. How tall is the lighthouse?

Exam Strategy Reminder:

1. Draw a diagram

2. Label known values

3. Identify the correct trigonometric ratio (TOA = Tangent = Opposite/Adjacent)

4. Set up your equation

5. Check that your calculator is in degree mode

6. Solve and round appropriately

Check out the tutorial here.