# 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 is an example of a right triangle. An angle of the triangle is labelled θ. The three sides of the right triangle are also given names (the adjacent, opposite and hyptonuse). Trigonometry studies how the lengths of the sides relate to each other and to the angle:

# Where Does the Word Trigonometry Come From?

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

# Right Triangles and Triangles

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

Note: Trigonometry usually studies right triangles, but can apply to all triangles, since any triangle can be made by putting two right triangles together:

# The Parts of a Right Triangle

The parts of a right triangle are (if we label an angle θ):
• A right triangle has one right angle (an angle of 90°). It is shown by drawing a small square.

• The longest side (opposite the right angle) is called the hypotenuse.

• The side next to the angle is called the adjacent:

• The side opposite the angle is called the opposite:

• Note: The adjacent and opposite only exist in relation to an angle. The same side can be an adjacent or an opposite depending on which angle they refer to.

# Sine, Cosine and Tangent

The sine function, cosine function and tangent function relate the angle of a right triangle to ratios of two of its sides.
• The sine function relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse.

The image below shows what we mean by a right triangle with an angle (labelled θ) and the three sides labelled. The sine function takes the angle. It is written sin θ.

• The cosine function relates the angle of a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse.

The image below shows what we mean by a right triangle with an angle (labelled θ) and the three sides labelled. The cosine function takes the angle. It is written cos θ.

• The tangent function relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side.

The image below shows what we mean by a right triangle with an angle (labelled θ) and the three sides labelled. The tangent function takes the angle. It is written tan θ.

# Using Trigonometry to Find Unknown Side Lengths in a Right Triangle

The trigonometric functions (sine, cosine and tangent) relate an angle to two of the sides of a right triangle. As long as we know an angle and a length of a side, we can always calculate the lengths of the other sides.

Knowing which trigonometric function to use can be tricky.

Out of the adjacent, the opposite and the hypotenuse; find out which length you know and which you are trying to work out.

Then use SOH CAH TOA to find which trigonometric function relates them:

# Using the Sine Function to Find Unknown Side Lengths in a Right Triangle

The sine function relates a given angle to the opposite side and hypotenuse of a right triangle. When you know two of these three, you can find the third.
• Use the sine function to find the length of the opposite side when the angle and the length of the hypotenuse are known.

• Use the sine function to find the length of the hypotenuse side when the angle and the length of the opposite are known.

# Using the Cosine Function to Find Unknown Side Lengths in a Right Triangle

The cosine function relates a given angle to the adjacent side and hypotenuse of a right triangle. When you know two of these three, you can find the third.
• Use the cosine function to find the length of the adjacent side when the angle and the length of the hypotenuse are known.

• Use the cosine function to find the length of the hypotenuse side when the angle and the length of the adjacent are known.

# Using the Tangent Function to Find Unknown Side Lengths in a Right Triangle

The tangent function relates a given angle to the opposite side and adjacent side of a right triangle. When you know two of these three, you can find the third.
• Use the tangent function to find the length of the opposite side when the angle and the length of the adjacent are known.

• Use the tangent function to find the length of the adjacent side when the angle and the length of the opposite is known.

# The Inverse Trigonometric Functions: Inverse Sine, Inverse Cosine and Inverse Tangent

The trigonometric functions (sine, cosine and tangent) take an angle of a right triangle as an input and give the ratio of two of the sides as an output:

The inverse trigonometric functions (inverse sine, inverse cosine and inverse tangent) do the opposite. They take a ratio of two of the sides as an input and give the angle as an output.

Inverse sine is labelled as sin-1, inverse cosine is labelled as cos-1 and inverse tangent is labelled as tan-1:

# Using Trigonometry to Find Unknown Angles in a Right Triangle

The inverse trigonometric functions can be used to find an unknown angle in a right triangle. As long as we know the lengths of two of the sides,we can always calculate the angle.
• Use the inverse sine function to find the angle when the length of the opposite side and the length of the hypotenuse are known.

• Use the inverse cosine function to find the angle when the length of the adjacent side and the length of the hypotenuse are known.

• Use the inverse tangent function to find the angle when the length of the opposite side and the length of the adjacent side are known.

Read more about using the inverse sine function to find the angle
Read more about using the inverse cosine function to find the angle
Read more about using the inverse tangent function to find the angle

# Trigonometry Lessons

Here is a handy list of our trigonometry lessons. The lessons and tests on these pages are designed to take a beginner through the basics of trigonometry. By the end of this curriculum, students will be solving some pretty daunting-looking trigonometry problems by themselves.

# Right Triangles

Understanding right triangles
Understanding right angles
Understanding the hypotenuse
Understanding the opposite

# The Sine, Cosine and Tangent Functions

Understanding the sine function
Understanding the cosine function
Understanding the tangent function

# Finding Unknown Lengths in a Right Triangle

Using the sine function to find the opposite
Using the sine function to find the hypotenuse
Using the cosine function to find the adjacent
Using the cosine function to find the hypotenuse
Using the tangent function to find the opposite
Using the tangent function to find the adjacent

# Finding Unknown Angles in a Right Triangle

Using the (inverse) sine function to find the angle
Using the (inverse) cosine function to find the angle
Using the (inverse) tangent function to find the angle