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."

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 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.
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 sine function is the ratio of the opposite to the hypotenuse.
- The cosine function is the ratio of the adjacent to the hypotenuse.
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:

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trigonometryfinding the hypotenuse using the cosine functionfinding the angle using the tangent functionfinding the opposite using the adjacent function
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