Tangent (tan) function  Trigonometry
(See also Tangent to a circle).
In a
right triangle,
the tangent of an angle is the length of the opposite side divided by the length of the
adjacent side.
Try this
Drag any
vertex of the triangle and see how the tangent of A and C are calculated.
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'.
Often remembered as "TOA"  meaning
Tangent is
Opposite over
Adjacent.
See SOH CAH TOA.
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 C is 0.5776 " or
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 tan 60° 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
Large and negative angles
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 Functions of large and negative angles.
When used this way we can also graph the tangent function. See Graphing the tangent function.
The derivative of tan(x)
In calculus, the derivative of tan(x) is sec^{2}(x).
This means that at any value of x, the rate of change or slope of tan(x) is sec^{2}(x).
For more on this see
Derivatives of trigonometric functions together with the derivatives of other trig functions.
See also the Calculus Table of Contents.
Other trigonometry topics
Angles
Trigonometric functions
Solving trigonometry problems
Calculus
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