Leaving Cert Trigonometry (Higher Level): Sine Rule, Cosine Rule & Identities

Trigonometry covers the sine and cosine rules, trig identities, the unit circle and radian measure. It is examined on Paper 2 and overlaps with geometry.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

The sine rule and cosine rule solve non-right-angled triangles.
The unit circle defines sin, cos and tan for all angles.
Trig identities are provided in the formula and tables booklet.
Trigonometry is examined on Paper 2.

Trigonometry explained

What is trigonometry?

Trigonometry is how we use maths to solve equations related to triangles.

We can solve for different sides and angles using many different formulae and functions. The most common trigonometric functions are sin, cos and tan and we will learn how to both use these for solving equations and how they act when graphed out as a function.

Pythagoras’ Theorem

Pythagoras’ theorem is one of the foundations of trigonometry. It applies to right-angled triangles. We use Pythagoras theorem only when given two sides of a right angled triangle and we want to solve for the third

If a right-angled triangle has sides of length $a$ , $b$ (the shorter sides) and hypotenuse of length $c$ , then:

$a^2 + b^2 = c^2$

Explanation:

The hypotenuse is always opposite the right angle and is the longest side.

This theorem links the lengths of the three sides of a right-angled triangle.

It is often used to find a missing side length.

Example

A right-angled triangle has legs of length $6$ and $8$ . Find the length of the hypotenuse.

$c^2 = 6^2 + 8^2 = 36 + 64 = 100$

$c = \sqrt{100} = 10$

So the hypotenuse is $10$ units long.

This is an introduction. The full Trigonometry lessons, worked solutions and practice questions are available to members. Start your free trial.

Key formulas

Name	Formula	Description
Sine Rule	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$	Relates sides to opposite angles in any triangle
Cosine Rule (Find Side)	$a^2 = b^2 + c^2 - 2bc\cos A$	Finds a side when two sides and included angle known
Cosine Rule (Find Angle)	$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$	Finds an angle when all three sides known
Area of Triangle	$\text{Area} = \frac{1}{2}ab\sin C$	Area using two sides and included angle
Pythagorean Identity	$\sin^2\theta + \cos^2\theta = 1$	Fundamental trigonometric identity
Tan Identity	$\tan\theta = \frac{\sin\theta}{\cos\theta}$	Definition of tangent in terms of sine and cosine

Worked examples

Worked example 1

If $\sin$ $\theta$ = $\frac{4}{5}$ and $\theta$ is in the first quadrant, find $\cos$ $\theta$ .

Using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ , we have $\cos^2\theta = 1 - \sin^2\theta = 1 - (\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25}$ . Since $\theta$ is in the first quadrant, $\cos\theta$ is positive, so $\cos\theta = \frac{3}{5}$ .

Answer:

\frac{3}{5}

Worked example 2

If $\sin\theta = \frac{3}{5}$ and $\theta$ is in the first quadrant, find $\cos\theta$ .

Using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ , we have $\cos^2\theta = 1 - \sin^2\theta = 1 - (\frac{3}{5})^2 = 1 - \frac{9}{25} = \frac{16}{25}$ . Since $\theta$ is in the first quadrant, $\cos\theta$ is positive, so $\cos\theta = \frac{4}{5}$ .

Answer:

\frac{4}{5}

Where students lose marks

Using the sine rule when the cosine rule is needed (and vice versa).
Leaving the calculator in degrees when the question is in radians.
Missing the second valid angle in the ambiguous case of the sine rule.

Frequently asked questions

When do I use the sine rule vs the cosine rule?

Use the cosine rule when you know two sides and the angle between them, or all three sides. Use the sine rule for most other cases involving a matching side and opposite angle.

What is the unit circle?

The unit circle is a circle of radius 1 used to define sine, cosine and tangent for every angle, including angles beyond 90 degrees.

Is trigonometry on Paper 1 or Paper 2?

Trigonometry is examined on Paper 2.

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