Tangent

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, or the slope of a line extending out in the direction of the said angle.

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 $\frac{1}{x^2+1}$