Secant (trigonometry)

Secant (trigonometry), one of the fundamental ratios of trigonometry. A ratio is a proportional relationship between two numbers calculated by dividing one number by the other. Secant embodies the relationship between the lengths of the sides of a right triangle (a triangle with a 90° angle) and the magnitudes of the angles. This relationship means that varying one value, such as the length of a side, requires another value, such as the magnitude of an angle, to change in a predictable way.

The secant, usually abbreviated sec, of one of the acute (less than 90°) angles of a right triangle is equal to the length of the longest side, called the hypotenuse, divided by the length of the side adjacent to the acute angle: .

Secant is also defined for angles greater than 90° using right triangles inscribed in a circle centered at the point (0,0) on the xy axis:

A line drawn from the circle’s center to any point on the circle makes an angle, θ, with the x axis. The secant of θ is equal to the length of the line connecting the point to the circle’s center divided by the horizontal distance of the point from the y axis. Secant smoothly increases in numerical value from negative infinity to -1 as θ increases from 90° to 180° and then decreases again, approaching negative infinity as θ goes from 180° to 270°. The function is discontinuous at 270°, flipping from negative infinity to positive infinity and then decreasing to 1 as θ moves from 270° to 360°.

Cosine is secant’s reciprocal function. The cosine, usually abbreviated cos, of an acute angle of a right triangle is equal to the length of the triangle’s hypotenuse divided by the length of the side adjacent to the chosen acute angle: .

Sine, Cosecant, Tangent, Cotangent.