The Lesson

The tangent function relates a given angle to the opposite side and adjacent side of a right triangle. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.

Dictionary Definition

The Merriam-Webster dictionary defines the tangent function as "the trigonometric function that for an acute angle is the ratio between the leg opposite to the angle when it is considered part of a right triangle and the leg adjacent."
The tangent of an angle is given by the formula below:

In this formula, tan denotes the tangent function, θ is an angle of a right triangle, the opposite is the length of the side opposite the angle and the adjacent is the length of the side adjacent the angle. The image below shows what we mean:

A Real Example of the Tangent Function

It is easier to understand the tangent function with an example.

Question

Find tan 45° using the right triangle shown below.

Step-by-Step:

1

Start with the formula:
tan θ = opposite / adjacent
Don't forget: / means ÷

2

Substitute the angle θ, the length of the opposite and the length of the adjacent into the formula. In our example, θ = 45°, the opposite is 4 cm and the adjacent is 4 cm.

tan (45°) = 4 / 4

tan (45°) = 4 ÷ 4

tan (45°) = 1

Answer:

tan 45° = 1.

The Graph of the Tangent Function

The tangent function can be plotted on a graph.

The tangent function is not defined at 90°, 270° (and any amount of 180° added or subtracted from these angles). This is seen on the graph as vertical red dashed lines, which the tangent function approaches but never touches. These lines are called asymptotes (see Note.) Find the angle along the horizontal axis, then go up until you reach the tangent graph. Go across and read the value of tan θ from the vertical axis. We can see from the graph above that tan 45° = 1.

The Tangent Function and the Unit Circle

The tangent function can be related to a unit circle, which is a circle with a radius of 1 that is centred at the origin in the Cartesian coordinate system. For a point at any angle θ, tan θ is given by the ratio of the y-coordinate to the x-coordinate of the point.

Lesson Slides

The slider below gives more information about the tangent function.

Interactive Widget

Here is an interactive widget to help you learn about the tangent function on a right triangle.

Trigonometry and Right Angles

The tangent function is a function in trigonometry (called a trigonometric function). The word trigonometry comes from the Greek words 'trigonon' ("triangle") and 'metron' ("measure"). Trigonometry is the branch of mathematics that studies the relationships between the sides and the angles of right triangles. When an angle is defined in a right triangle, the three sides can be defined.

  • The side next to the angle is called the adjacent.
  • The side opposite the angle is called the opposite.
  • The longest side is called the hypotenuse.

Other Trigonometric Functions

The tangent function is only one of the trigonometric functions:

The Tangent Function and Asymptotes

It follows from the definition of the tangent function, and of the sine and cosine functions, that the tangent function can be expressed as:

When cos θ = 0 (at θ = 90°, 270°, ...), sin θ is being divided by 0. When any number is divided by 0, the result is undefined. At these values of θ, tan θ is undefined. On the graph, these appear as vertical asymptotes.

Either side of these values of θ, cos θ is very close to 0. It is a very small positive or negative number. At these values of θ, sin θ is being divided by a very small positive or negative number. The result is a very large positive or negative number. Either side of the asymptotes, tan θ approaches plus or minus infinity.