Trigonometry: sine, cosine and tangent

Years 9 to 11

In a right-angled triangle, sin, cos and tan link an angle to the ratio of two sides.

A right-angled triangle with the sides named from θ: sin θ = opp ÷ hyp, cos θ = adj ÷ hyp, tan θ = opp ÷ adj.

For learners

Pick one angle in a right-angled triangle and call it θ. The side opposite it is the opposite. The side next to it is the adjacent. The longest side is the hypotenuse.

Sine is opposite ÷ hypotenuse. Cosine is adjacent ÷ hypotenuse. Tangent is opposite ÷ adjacent.

To remember: SOH CAH TOA. Pick the two sides you have, then the ratio that uses them.

A worked example

In a right-angled triangle the hypotenuse is 10 cm and the angle θ is 30°. Find the opposite side.

We have the hypotenuse and want the opposite, so use sine.
$sin 30° = \frac{opp}{10}$ , and $sin 30° = 0.5$ .
Opposite $= 0.5 \times 10 = 5$ cm.

For grown-ups

Label the sides from the angle you are working with first. Then circle the two sides involved, and choose sin, cos or tan. Make sure the calculator is in degree mode. To find an angle, use the inverse function, such as sin⁻¹.

Common mistakes

Labelling opposite and adjacent from the wrong angle.
Calculator set to radians instead of degrees.
Using trigonometry on a triangle with no right angle (use the sine or cosine rule instead).

Why it matters

Surveyors, pilots, builders and game designers use trigonometry to work out heights and distances they cannot measure directly.

Topics in quarter that use this

Right-angled trigonometry
Exact trigonometric values
Sine and cosine rules
Graphs of sine and cosine
Trigonometric identities