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Tangent is a trigonometric ratio. In a right triangle with an angle \theta ,

\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}

\text{Opposite} is the side of the triangle not part of angle \theta , and \text{adjacent} is the side that is.

Proof

The proof that \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} is one of the most simple proofs around. We know that \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} and \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}. Dividing them gives \frac{\text{opposite}}{\text{adjacent}} which is also equal to \tan(\theta).

Properties

The tangent of an angle is the sine of said angle divided by cosine.

The reciprocal of tangent is cotangent (abbreviated as \cot), while its inverse is \arctan or \tan^{-1} . Note that tangent is not being raised to the power of -1; this is an inverse function, not a reciprocal.

The derivative of \tan(x) is \sec^2(x) , while its antiderivative is -\ln\big(|\cos(x)|\big) .

The derivative of \arctan(x) is \frac{1}{x^2+1}

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