1.5933678
0.62760149
-28.95280245
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1.5933678
0.62760149
-28.95280245
rad
0
The Cosecant Calculator computes the cosecant of any angle in degrees or radians. Cosecant (abbreviated csc) is the reciprocal of sine and one of the three reciprocal trigonometric functions used in advanced mathematics, physics, and engineering.
Cosecant is defined as:
$$\csc\theta = \frac{1}{\sin\theta} = \frac{\text{hypotenuse}}{\text{opposite}}$$
Since cosecant is the reciprocal of sine, it is undefined wherever sine equals zero — at $$\theta = 0°, 180°, 360°,$$ and all integer multiples of $$180°$$. At these angles, the function has vertical asymptotes. The absolute value of cosecant is always greater than or equal to 1, because $$|\sin\theta| \leq 1$$ implies $$|\csc\theta| \geq 1$$.
The range of cosecant is $$(-\infty, -1] \cup [1, +\infty)$$. It never takes values between $$-1$$ and $$1$$. Cosecant is positive in Quadrants I and II (where sine is positive) and negative in Quadrants III and IV.
Key exact values include:
$$\csc 30° = 2, \quad \csc 45° = \sqrt{2}, \quad \csc 60° = \frac{2}{\sqrt{3}}, \quad \csc 90° = 1$$
Like sine, cosecant is an odd function: $$\csc(-\theta) = -\csc\theta$$, and it has a period of $$2\pi$$ radians (360°).
Cosecant satisfies the Pythagorean identity:
$$1 + \cot^2\theta = \csc^2\theta$$
This identity is derived by dividing the fundamental identity $$\sin^2\theta + \cos^2\theta = 1$$ by $$\sin^2\theta$$. It is particularly useful in integration and when simplifying trigonometric expressions.
In calculus, the derivative of cosecant is $$\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)$$, and its integral is $$-\ln|\csc(x) + \cot(x)| + C$$. These formulas appear frequently in integration problems involving rational expressions of trigonometric functions.
While cosecant is less commonly used than sine, cosine, or tangent in elementary applications, it plays a significant role in wave optics (particularly in diffraction patterns), complex analysis (where it connects to the Weierstrass product), and hyperbolic geometry. In navigation and surveying, the cosecant function can simplify formulas for computing distances and bearings.
This calculator also displays the corresponding sine value, allowing you to verify the reciprocal relationship directly. If the sine is close to zero, the cosecant grows extremely large, illustrating the behavior near the asymptotes.
The calculator converts the angle to radians if needed, computes $$\sin(\theta)$$, and then takes its reciprocal: $$\csc\theta = \frac{1}{\sin\theta}$$. If $$|\sin\theta| < 10^{-10}$$, the result is undefined (NaN) to avoid division by near-zero values.
Cosecant is always $$\geq 1$$ or $$\leq -1$$, never between. An undefined result means sine is zero at that angle (0°, 180°, 360°). The accompanying sine value lets you verify the reciprocal relationship: $$\csc\theta \times \sin\theta = 1$$.
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csc(30°) = 1/sin(30°) = 1/0.5 = 2. This is one of the standard exact values commonly used in trigonometry courses.
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At 210° (Quadrant III), sin(210°) = −0.5, so csc(210°) = −2. The negative sign reflects that sine is negative in Quadrant III.
Since $$\csc\theta = \frac{1}{\sin\theta}$$, the function is undefined where $$\sin\theta = 0$$. Sine equals zero at $$0°, 180°, 360°,$$ and all integer multiples of $$180°$$. At these points, cosecant has vertical asymptotes and the function value approaches $$\pm\infty$$.
Cosecant can take any value in $$(-\infty, -1] \cup [1, +\infty)$$. It never falls between $$-1$$ and $$1$$ because its reciprocal (sine) has magnitude at most 1. The minimum absolute value of 1 occurs when $$\sin\theta = \pm 1$$ (at 90° and 270°).
Cosecant is the reciprocal of sine: $$\csc\theta = \frac{1}{\sin\theta}$$. It satisfies the identity $$1 + \cot^2\theta = \csc^2\theta$$. It can also be expressed as $$\csc\theta = \frac{\sec\theta}{\tan\theta}$$ or $$\csc\theta = \sec(90° - \theta)$$.
Yes. Cosecant appears in physics formulas for diffraction patterns, in engineering calculations involving resonance and wave interference, and in navigation formulas. The cosecant rule for spherical triangles is used in astronomy and global positioning calculations.
The cosecant graph consists of U-shaped curves separated by vertical asymptotes at every multiple of $$180°$$. Between 0° and 180°, the curve opens upward with minimum value 1 at 90°. Between 180° and 360°, it opens downward with maximum value $$-1$$ at 270°. The pattern repeats with period 360°.
$$\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)$$. This derivative is negative in the first quadrant (where both csc and cot are positive), meaning cosecant is decreasing there. The integral is $$\int \csc(x)\,dx = -\ln|\csc(x) + \cot(x)| + C$$.
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