secant

secant, one of the six trigonometric functions, which, in a right triangle ABC, for an angle A, issec A = length of hypotenuse/length of side adjacent angle A.(The other five trigonometric functions are sine [sin], cosine [cos], tangent [tan], cosecant [csc], and cotangent [cot].)

From the definition of the tangent of angle A,tan A = length of side opposite to angle A/length of side adjacent to angle A, and the Pythagorean theorem, one has the useful identitytan² A + 1 = sec² A.

The reciprocal of the secant is the cosine: 1/sec A = cos A.

When A is expressed in radians, the secant function has a period of 2π. The function has a value of 1 at 0 and −1 at π. At π/2 the function diverges to positive infinity when approaching that number from x < π/2 and diverges to negative infinity when approaching that number from x > π/2. Similar behaviour occurs at 3π/2, but the function diverges to negative infinity when approaching that number from x < π/2 and diverges to positive infinity when approaching that number from x > π/2. Also, sec (−A) = sec A.

With respect to x, the derivative of sec x is sec x tan x, and the indefinite integral of sec x is ∫sec x dx = ln |sec x + tan x|,where ln is the natural logarithm.