tangent

tangent, one of the six trigonometric functions, which, in a right triangle ABC, for an angle A, is$tan A = \frac{length of side opposite angle A }{length of side adjacent to angle A}$.The other five trigonometric functions are sine (sin), cosine (cos), secant (sec), cosecant (csc), and cotangent (cot).

Relationship to sine and cosine

From the definition of the sine and the cosine of angle A$\sin A = \frac{length of side opposite angle A }{length of hypotenuse}$ and $\cos A = \frac{length of side adjacent to angle A }{length of hypotenuse}$one obtains$\tan A = \frac{\sin A }{\cos A}$.

Identities

A number of algebraic identities involving the tangent function are useful in trigonometry, particularly for simplifying expressions, solving equations, and analyzing angle relationships.

Pythagorean identity: From the definition of the secant of angle A,$sec A = \frac{length of hypotenuse}{length of side adjacent to angle A}$

and the Pythagorean theorem, one obtains the identity${\tan }^2 A + 1 = se{c}^2 A$.

Half-angle formula:$tan \left(\frac{A}{2}\right) = \frac{sin A}{1+cos A}$ and $tan \left(\frac{A}{2}\right) = \frac{1-cos A}{sin A}$

Double-angle formula:$tan 2A = \frac{2 tan A}{1 - ta{n}^2 A}$

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Addition formula:$tan (A+B) = \frac{tan A + tan B}{1 - tan A tan B}$

Subtraction formula:$tan (A-B) = \frac{tan A - tan B}{1 + tan A tan B}$

Geometric and graphical characteristics

If a circle with radius 1 has its center at the origin (0,0) and a line is drawn through the origin with an angle A with respect to the x-axis, the tangent is the slope of the line. It can also be interpreted as:$\tan A = \frac{y}{x}$where (x, y) is the point where the terminal side of angle A intersects the unit circle. When A is expressed in radians, the tangent function has a period of π. It is an odd function, satisfying the identity tan (−A) = −tan A, and it has vertical asymptotes wherever the cosine of the angle equals zero—that is, at $A = \frac{\pi }{2} + n\pi$,where n is any integer.

Reciprocal function, derivative, and integral

The reciprocal of the tangent is the cotangent:$\frac{1}{tan A} = cot A$.

With respect to x, the derivative of tan x is sec² x, and the indefinite integral of tan x is −ln |cos x|, where ln is the natural logarithm.