Cosecant (CSC) Calculator

Calculate csc(θ) = 1/sin(θ) and view all six trigonometric function values for any angle.

Enter Angle

Result

csc(θ)
2
csc(30°) = 1/sin(30°)
sin(θ)0.5
cos(θ)0.866025
tan(θ)0.577350
csc(θ)2
sec(θ)1.154701
cot(θ)1.732051
Angle in degrees30°
Angle in radians0.523599

Step-by-Step Solution

csc(30°) = 1/sin(30°) = 1/0.5 = 2

Understanding the Cosecant Function

The cosecant function (csc) is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function: csc(θ) = 1/sin(θ). In a right triangle, the cosecant of an angle is the ratio of the hypotenuse to the side opposite the angle.

The cosecant function is undefined wherever sine equals zero, which occurs at integer multiples of π (0, π, 2π, ...) or equivalently at 0°, 180°, 360°, and so on. At these points, the function has vertical asymptotes. The range of the cosecant function is (-infinity, -1] union [1, infinity) -- it never takes values between -1 and 1.

Trigonometric Function Relationships

Cosecant (csc)

The reciprocal of sine. Defined for all angles where sin is non-zero.

csc(θ) = 1/sin(θ) = hyp/opp

Secant (sec)

The reciprocal of cosine. Undefined where cos = 0.

sec(θ) = 1/cos(θ) = hyp/adj

Cotangent (cot)

The reciprocal of tangent, or cos/sin.

cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)

Pythagorean Identity

A fundamental identity relating csc and cot.

1 + cot^2(θ) = csc^2(θ)

Key Values of CSC

Common exact values for reference angles.

csc(30°) = 2, csc(45°) = sqrt(2), csc(60°) = 2sqrt(3)/3

Period

The cosecant function has a period of 2π (360°).

csc(θ + 2π) = csc(θ)

Graph and Properties of CSC

The graph of y = csc(x) consists of a series of U-shaped curves opening upward and downward, alternating between intervals. Between each pair of consecutive asymptotes, the function reaches a minimum of 1 or a maximum of -1. The function is odd, meaning csc(-x) = -csc(x), so the graph has rotational symmetry about the origin.

Domain and Range

  • Domain: All real numbers except where sin(θ) = 0, i.e., θ is not equal to nπ for any integer n.
  • Range: (-infinity, -1] union [1, infinity). The function never takes values between -1 and 1.
  • Period: 2π radians (360°).
  • Symmetry: Odd function, so csc(-θ) = -csc(θ).

Applications of the Cosecant Function

While sine, cosine, and tangent are the most commonly used trigonometric functions, the cosecant appears in many mathematical and scientific contexts. It shows up in calculus (integration formulas), wave mechanics, optics (Fresnel equations), and electromagnetic theory. The cosecant is also used in navigation, surveying, and acoustic engineering.

Common Special Values

  • csc(30°) = csc(π/6) = 2
  • csc(45°) = csc(π/4) = √2 (approximately 1.41421)
  • csc(60°) = csc(π/3) = 2√3/3 (approximately 1.15470)
  • csc(90°) = csc(π/2) = 1
  • csc(0°) and csc(180°) are undefined (division by zero)

Tips for Using CSC

  • Always check whether the angle makes sin(θ) = 0 before computing csc(θ).
  • Remember that csc is always greater than or equal to 1 in absolute value.
  • Use the identity 1 + cot^2(θ) = csc^2(θ) to find csc from cot.
  • On a calculator, compute csc as 1 divided by sin of the angle.