Trigonometry

What Is Trigonometry?

Trigonometry is a branch of mathematics that studies the angles and lengths of the sides of triangles. Trigonometry particularly deals with functions which relate the angle of a right triangle with the lengths of its sides: the sine function, cosine function and tangent function.

Dictionary Definition

The Oxford English Dictionary defines trigonometry as "that branch of mathematics which deals with the measurement of the sides and angles of triangles, particularly with certain functions of their angles or of angles in general (the sine, cosine and tangent)."

Here are some examples from trigonometry. We might be interested in finding the angle or the length of a side of a right triangle. We might want to plot a function found from the ratio of the lengths of two sides of a right triangle. Trigonometry allows us to do this.

Where Does the Word Trigonometry Come From?

Trigonometry comes from the Greek words 'trigonon' ("triangle") and 'metron' ("measure").

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning. A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Right Triangle

Right triangles (in American English) or right-angled triangles (in English English) are triangles containing one angle of 90° (called a right angle). In this mini-curriculum, you will learn about right triangles and their sides.

Right Triangle
A right triangle (or a right-angled triangle) is a triangle containing an angle of 90° (called a right angle).

Adjacent
A right triangle has three sides: one side is the longest and two are shorter sides.
If we define an angle in the right triangle, the adjacent is the shorter side next to (adjacent to) that angle.

Opposite
A right triangle has three sides: one side is the longest and two are shorter sides.
If we define an angle in the right triangle, the opposite is the shorter side opposite that angle.

Hypotenuse
The hypotenuse is the longest side in a right triangle. It is the side opposite the 90° angle.

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Read more about right triangles
Read more about the adjacent
Read more about the opposite
Read more about the hypotenuse

Pythagoras' Theorem

Pythagoras' theorem concerns the relationship between the lengths of the three sides of a right triangle. In this mini-curriculum, you will learn about Pythagoras' theorem and how to use it.

Pythagoras' theorem
Pythagoras’ theorem (or the Pythagorean theorem) states that: "The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides."
It is easier to remember Pythagoras' theorem as a formula:

In this right triangle, the hypotenuse is c and the two shorter sides are a and b. Hence if the square of the hypotenuse is equal to the sum of squares of the other two sides: c² = a² + b²

Find the Hypotenuse Using Pythagoras' Theorem
To find the hypotenuse using Pythagoras' theorem, rearrange the formula and substitute in the known lengths:

Find a Shorter Side Using Pythagoras' Theorem
To find a shorter side using Pythagoras' theorem, rearrange the formula and substitute in the known lengths:

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Read more about Pythagoras' theorem
Read more about how to find the hypotenuse using Pythagoras' theorem
Read more about how to find a shorter side using Pythagoras' theorem

Sine Function

The sine function is a trigonometric function. The sine of the angle in a right triangle is equal to the ratio of the length of the opposite side to the length of the hypotenuse. In this mini-curriculum, you will learn about the sine function and how it can be used to find angles and lengths of sides in a right triangle.

Sine Function
The sine function relates a given angle to the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
Use the Sine Function to Find the Angle
The sine function can be used to find the angle when the opposite and hypotenuse are known.

Use the Sine Function to Find the Opposite
The sine function can be used to find the opposite when the angle and hypotenuse are known.

Use the Sine Function to Find the Hypotenuse
The sine function can be used to find the hypotenuse when the angle and opposite are known.

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Read more about the sine function
Read more about how to find the angle using the sine function
Read more about how to find the opposite using the sine function
Read more about how to find the hypotenuse using the sine function

Cosine Function

The cosine function is a trigonometric function. The cosine of the angle in a right triangle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse. In this mini-curriculum, you will learn about the cosine function and how it can be used to find angles and lengths of sides in a right triangle.

Cosine Function
The cosine function relates a given angle to the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

Use the Cosine Function to Find the Angle
The cosine function can be used to find the angle when the adjacent and hypotenuse are known.

Use the Cosine Function to Find the Adjacent
The cosine function can be used to find the adjacent when the angle and hypotenuse are known.

Use the Cosine Function to Find the Hypotenuse
The cosine function can be used to find the hypotenuse when the angle and adjacent are known.

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Read more about the cosine function
Read more about how to find the angle using the cosine function
Read more about how to find the adjacent using the cosine function
Read more about how to find the hypotenuse using the cosine function

Tangent Function

The tangent function is a trigonometric function. The tangent of the angle in a right triangle is equal to the ratio of the length of the opposite side to the length of the adjacent side. In this mini-curriculum, you will learn about the tangent function and how it can be used to find angles and lengths of sides in a right triangle.

Tangent Function
The tangent function relates a given angle to the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.

Use the Tangent Function to Find the Angle
The tangent function can be used to find the angle when the opposite and adjacent are known.

Use the Tangent Function to Find the Opposite
The tangent function can be used to find the opposite when the angle and adjacent are known.

Use the Tangent Function to Find the Adjacent
The tangent function can be used to find the adjacent when the angle and opposite are known.

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Read more about the tangent function
Read more about how to find the angle using the tangent function
Read more about how to find the opposite using the tangent function
Read more about how to find the adjacent using the tangent function

Area of a Triangle Using Trigonometry

Even though trigonometry mostly concerns right triangles, it can be used for any triangle. In this mini-curriculum, you will learn about finding the area of a triangle using trigonometry.

Area of a Triangle Using Trigonometry
The area of a triangle where a, b and c are the lengths of the sides and A, B and C are the angles is:

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Read more about how to find the area of a triangle using trigonometry