Cosecant (csc)

What is cosecant

Cosecant is the reciprocal of sine: csc α = 1 / sin α. Like secant, it is mostly useful in higher mathematics.

Domain and properties

The domain is all real numbers except k·180°. The range is (−∞, −1] ∪ [1, +∞). The period is 360° and the function is odd.

Notable values

csc 30° = 2, csc 90° = 1

Where cosecant is used in practice

• Integral calculus and solving differential equations.
• Wave physics when describing amplitudes.
• Historically in astronomical calculations.

The unit circle and graphs

The best way to understand trigonometric functions is the unit circle – a circle of radius 1 centred at the origin. Moving along it by an angle α, the coordinates of the point are exactly (cos α, sin α). This gives the fundamental identity sin²α + cos²α = 1 (the Pythagorean theorem for that point). The graph of the function is obtained by plotting the function value against the angle – producing the characteristic wave.

Relationships between the functions

tan α = sin α / cos α  •  cot α = 1 / tan α  •  sin²α + cos²α = 1