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Calculate the cosecant of any angle. Cosecant is the reciprocal of sine: csc(θ) = 1/sin(θ).
| Degrees | Radians | Exact Value | Decimal |
|---|---|---|---|
| 0° | 0 | undefined | — |
| 30° | π/6 | 2 | 2.0000 |
| 45° | π/4 | √2 | 1.4142 |
| 60° | π/3 | 2√3/3 | 1.1547 |
| 90° | π/2 | 1 | 1.0000 |
| 120° | 2π/3 | 2√3/3 | 1.1547 |
| 135° | 3π/4 | √2 | 1.4142 |
| 150° | 5π/6 | 2 | 2.0000 |
| 180° | π | undefined | — |
| 270° | 3π/2 | -1 | -1.0000 |
All θ except 0°, 180°, 360°, ...
Where sin(θ) ≠ 0
(-∞, -1] ∪ [1, ∞)
|csc(θ)| ≥ 1
2π radians (360°)
csc(θ + 2π) = csc(θ)
Odd function: csc(-θ) = -csc(θ)
Origin symmetry
Cosecant (abbreviated csc) is one of the six trigonometric functions. It is defined as the reciprocal of the sine function: csc(θ) = 1/sin(θ). In a right triangle, cosecant equals the ratio of the hypotenuse to the opposite side.
Since cosecant is the reciprocal of sine, it inherits many of sine's properties but with key differences. The range restriction (|csc(θ)| ≥ 1) comes from the fact that |sin(θ)| ≤ 1, so its reciprocal must have absolute value at least 1. Cosecant is undefined wherever sine equals zero.
Cosecant is the reciprocal of sine. It equals 1 divided by the sine of the angle, or in a right triangle, hypotenuse divided by opposite side.
Since |sin(θ)| ≤ 1, and csc = 1/sin, the absolute value of cosecant must be ≥ 1 (except at undefined points).
Cosecant is undefined when sin(θ) = 0, which occurs at 0°, 180°, 360°, and all integer multiples of 180°.
The range of cosecant is (-∞, -1] ∪ [1, ∞). The function never takes values between -1 and 1.
The period of cosecant is 2π radians (360°), the same as sine. This means csc(θ + 2π) = csc(θ).
Cosecant equals the reciprocal of the y-coordinate on the unit circle (where that coordinate is non-zero).
Cosecant is an odd function: csc(-θ) = -csc(θ). This follows from sine being an odd function.
Cosecant appears in wave physics, optics, and solving triangle problems. It is less commonly used than sine but important in certain calculus applications.