Cosecant Calculator
Free Cosecant Calculator for trigonometry. Enter values to get step-by-step solutions with formulas and graphs. See charts, tables, and visual results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
csc(x) = 1 / sin(x) = hypotenuse / opposite
The cosecant of angle x equals the reciprocal of the sine of x. In a right triangle, it is the ratio of the hypotenuse to the side opposite the angle. The function is undefined when sin(x) = 0.
Worked Examples
Example 1: Cosecant of 30 Degrees
Problem:Calculate csc(30 degrees) and verify it using the reciprocal relationship with sine.
Solution:sin(30) = 0.5\ncsc(30) = 1 / sin(30) = 1 / 0.5 = 2.0\nVerification: sin(30) x csc(30) = 0.5 x 2.0 = 1.0 (confirmed)\nPythagorean check: csc2(30) - cot2(30) = 4 - 3 = 1 (confirmed)
Result:csc(30) = 2.0 | sin(30) = 0.5 | Reciprocal verified
Example 2: Cosecant of 210 Degrees
Problem:Find csc(210 degrees) and determine its sign based on the quadrant.
Solution:210 degrees is in the third quadrant (180-270).\nReference angle = 210 - 180 = 30 degrees.\nsin(210) = -sin(30) = -0.5\ncsc(210) = 1 / sin(210) = 1 / (-0.5) = -2.0\nCosecant is negative in Q3, consistent with sine being negative in Q3.
Result:csc(210) = -2.0 | Quadrant III (negative) | Reference angle = 30
Frequently Asked Questions
What is the cosecant function in trigonometry?
The cosecant function, written as csc(x) or cosec(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x), which in a right triangle equals the ratio of the hypotenuse to the side opposite the angle. The cosecant function is one of the six standard trigonometric functions, though it is used less frequently than sine and cosine. Its values are always greater than or equal to 1, or less than or equal to -1, meaning there is a gap in its range between -1 and 1 where no output values exist. Understanding cosecant is essential for working with trigonometric identities and solving certain types of equations.
When is cosecant undefined?
The cosecant function is undefined whenever the sine of the angle equals zero, because division by zero is mathematically undefined. This occurs at integer multiples of 180 degrees (or pi radians), specifically at 0, 180, 360, 540 degrees and so on, including their negative counterparts. At these angles, the opposite side of the right triangle has zero length, making the hypotenuse-to-opposite ratio impossible to compute. On a graph, the cosecant function has vertical asymptotes at each of these points. When working with the cosecant function, it is important to check whether your input angle falls on or near these values to avoid computational errors or meaningless results.
What is the range and domain of cosecant?
The domain of the cosecant function includes all real numbers except integer multiples of pi radians (or 180 degrees), where sine equals zero. The range of cosecant is all real numbers y such that y is less than or equal to -1 or y is greater than or equal to 1. In interval notation, the range is written as (-infinity, -1] union [1, +infinity). This means cosecant never takes values between -1 and 1. The function oscillates between these regions with a period of 360 degrees (2pi radians). In the first and second quadrants cosecant is positive, while in the third and fourth quadrants it is negative, following the same sign pattern as sine.
How is cosecant related to other trigonometric functions?
Cosecant is directly connected to all other trigonometric functions through fundamental identities. As the reciprocal of sine, csc(x) = 1/sin(x) is the most basic relationship. The Pythagorean identity for cosecant states that csc squared minus cot squared equals 1, analogous to the primary Pythagorean identity. Cosecant can be expressed in terms of cosine as csc(x) = 1/sqrt(1 - cos squared x), and in terms of tangent and secant through various derived identities. The cofunction identity csc(x) = sec(90 - x) connects cosecant to secant. These relationships are used extensively in calculus for integration and in physics for modeling wave phenomena and oscillations.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy