An angle fixes a ratio.

Name sides relative to the chosen angle: the hypotenuse is opposite the right angle, the opposite side faces the chosen angle, and the adjacent side touches it. The ratios stay the same for similar right triangles.

Use an inverse trig function to recover an angle from a ratio. For Year 10 triangle problems, check that the calculator is in degree mode. Sketch the situation, mark the right angle and label the known measurements before choosing a formula.

INTERACTIVE MODEL

Try it. Watch it change.

Side a control (a = 4 + control)1Side b control (b = 5 + control)0Angle C / degrees78

c² = a² + b² − 2ab cos C · area = 12.2268 square units

Sides and angle are calculated to scale. At 90°, the cosine term is zero and c is the hypotenuse.

Explore: Change the included angle. Set 90° to see Pythagoras as a special case of the cosine rule.

Sine

\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}

Tangent

\tan\theta=\frac{\text{opposite}}{\text{adjacent}}

WORKED EXAMPLE

A ladder of length 5 m makes a 60° angle with level ground. How high does it reach?

Height is opposite the angle; 5 m is the hypotenuse.
h/5 = sin(60°), so h = 5sin(60°) ≈ 4.33 m.

Assumed knowledge

Similar triangles and Pythagoras’ theorem.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use sine, cosine and tangent in right triangles.
Choose a ratio from the known and required sides.

Year 10 preparation ↗

READY TO TRY IT?

An angle fixes a ratio.

A ladder of length 5 m makes a 60° angle with level ground. How high does it reach?

Put the idea to work.