Sine, cosine and tangent: a ladder against a wall
Lean a ladder against a wall at different angles and see how sin, cos and tan compare its sides; find the safe angle.
Core concept
For an angle θ in a right-angled triangle, sin θ = opp ÷ hyp, cos θ = adj ÷ hyp and tan θ = opp ÷ adj; they depend only on the angle
Topic: Trigonometric ratios
A ladder leaning against a wall makes a right-angled triangle with the wall and the ground. Seen from the angle: The amount of turn between two lines that meet, measured in degrees (°). All key words θ at the foot of the ladder, the wall is the opposite side, the ground is the adjacent side and the ladder is the hypotenuse. Sine, cosine and tangent are the ratio: A way to compare two amounts, like 2 cups of rice to 3 cups of water (2 : 3). All key words of these sides: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent. A longer ladder at the same angle makes a bigger, similar triangle, so the ratios stay the same: they depend only on the angle. Builders say a ladder is safe when its foot is about a quarter of its height away from the wall, about 75°.
What you will learn
- Name the opposite, adjacent and hypotenuse sides for an angle.
- Find sin, cos and tan as ratios of sides.
- Use a ratio to find an unknown side, such as how high a ladder reaches.
Before you start
- Right-angled triangles and Pythagoras' theorem.
Materials
- A ladder or a ruler leaning on a box
Equipment
- A measuring tape and a protractor
Procedure
- Drag the top of the ladder up or down the wall to change the angle; tap − or + for the ladder's length.
- Read the sides and the three ratios.
- Record the ladder at 30°, at 45° and at 60°.
- Find a safe angle where the foot is about a quarter of the height out, and record it.
How this simulation is simplified
- The wall is vertical and the ground is level.