Tangent (tan) function - Trigonometry

In a right triangle, the tangent of an angle is the length of the opposite side divided by the length of the adjacent side.

The tangent function, along with sine and cosine, is one of the three most common trigonometric functions. In any right triangle, the tangent of an angle is the length of the opposite side (O) divided by the length of the adjacent side (A). In a formula, it is written simply as 'tan'.

As an example, let's say we want to find the tangent of angle C in the figure above (click 'reset' first). From the formula above we know that the tangent of an angle is the opposite side divided by the adjacent side. The opposite side is AB and has a length of 15. The adjacent side is BC with a length of 26. So we can write

This division on the calculator comes out to 0.577. So we can say "The tangent of 30° is 0.5776 " or

Calculator

Example - using tangent to find a side length

If we look at the general definition -

we see that there are three variables: the measure of the angle x, and the lengths of the two sides (Opposite and Adjacent). So if we have any two of them, we can find the third.

In the figure above, click 'reset'. Imagine we didn't know the length of the side BC. We know that the tangent of A (60°) is the opposite side (26) divided by the adjacent side AB - the one we are trying to find.

From our calculator we find that tan60 is 1.733, so we can write

Transposing:

which comes out to 26, which matches the figure above.

The inverse tangent function - arctan

For every trigonometry function such as tan, there is an inverse function that works in reverse. These inverse functions have the same name but with 'arc' in front. So the inverse of tan is arctan etc. When we see "arctan A", we interpret it as "the angle whose tangent is A"

tan 60 = 1.733	Means: The tangent of 60 degrees is 1.733
arctan 1.733 = 60	Means: The angle whose tangent is 1.733 is 60 degrees.

We use it when we know what the tangent of an angle is, and want to know the actual angle.
See also Arctangent definition and Inverse functions - trigonometry

Example - using arctan to find an angle

In the above figure, imagine we know the side lengths but need to find the measure of angle C.
We know that tan C = 0.577 (15 over 26) so we need to know the angle whose tangent is 0.577, or formally:

Using our calculator to look up arctan 0.577 we find it is 60°. On some calculators the arctan button may be labelled atan, or sometimes tan^-1.

Angles greater than 90°

In a right triangle, the two variable angles are always less than 90° (See Interior angles of a triangle). But we can in fact find the tangent of any angle, no matter how large, and also the tangent of negative angles. For more on this see tangent definition for any angle.

When used this way we can also graph the tangent function. See Graphing the tangent function.

Other trigonometry topics

Angles

• Angle definition, properties of angles
• Standard position on an angle
• Initial side of an angle
• Terminal side of an angle
• Quadrantal angles
• Coterminal angles

Trigonometric functions

• Introduction to the six trig functions
• SOH CAH TOA memory aid


• Sine function (sin) in right triangles
• Graphing the sine function
• Sine waves


• Cosine function (cos) in right triangles
• Graphing the cosine function


• Tangent function (tan) in right triangles
• Graphing the tangent function


• Cotangent function cot (in right triangles)
• Secant function sec (in right triangles)
• Cosecant function csc (in right triangles)

Solving trigonometry problems

• The general approach
• Finding slant distance along a slope or ramp
• Finding the angle of a slope or ramp