Cosecant Calculator
Calculate csc(x) = 1/sin(x) for any angle
Common Cosecant Values
0°
Undefined
30°
2
45°
√2 ≈ 1.414
60°
2√3/3 ≈ 1.155
90°
1
180°
Undefined
270°
-1
360°
Undefined
How to Use This Calculator
Enter the Angle
Input the angle value you want to calculate the cosecant for. You can use degrees or radians.
Select Unit
Choose whether your angle is in degrees (°) or radians (rad).
Calculate
Click "Calculate Cosecant" to get the result. Note that cosecant is undefined when sin(θ) = 0.
Review Results
View the cosecant value along with related trigonometric functions for the same angle.
Formula
csc(θ) = 1 / sin(θ)
Reciprocal of sine
Definition:
Cosecant is the reciprocal of the sine function. It's one of the six trigonometric functions and is often used in trigonometry and calculus.
Properties:
- Range: (-∞, -1] ∪ [1, ∞) - all real numbers except between -1 and 1
- Period: 360° (2π radians) - same as sine
- Undefined: When sin(θ) = 0 (at 0°, 180°, 360°, etc.)
- Odd Function: csc(-θ) = -csc(θ)
Right Triangle Definition:
csc(θ) = hypotenuse / opposite
About Cosecant Calculator
The Cosecant Calculator is a specialized tool for calculating cosecant values for any angle. Cosecant (csc) is the reciprocal of the sine function, meaning csc(θ) = 1/sin(θ). It's one of the six fundamental trigonometric functions and is widely used in advanced mathematics, physics, and engineering applications.
What is Cosecant?
Cosecant is the reciprocal trigonometric function of sine. In a right triangle, cosecant of an angle is the ratio of the hypotenuse to the opposite side. On the unit circle, cosecant represents the reciprocal of the y-coordinate (since sine is the y-coordinate).
When to Use This Calculator
- Trigonometric Calculations: Find cosecant values for angles in degrees or radians
- Physics Problems: Calculate values needed for wave equations and oscillations
- Engineering: Solve problems involving periodic functions and vibrations
- Mathematics: Simplify trigonometric expressions involving cosecant
- Verification: Verify manual calculations and check homework answers
Why Use Our Calculator?
- ✅ Instant Results: Get accurate cosecant values immediately
- ✅ Multiple Units: Supports both degrees and radians
- ✅ Related Functions: Shows sine, cosine, tangent, and secant for comparison
- ✅ Undefined Detection: Clearly indicates when cosecant is undefined
- ✅ Easy to Use: Simple interface for all skill levels
- ✅ 100% Free: No registration or payment required
- ✅ Mobile Friendly: Works on all devices
Important Notes
- Undefined Values: Cosecant is undefined when sin(θ) = 0, which occurs at 0°, 180°, 360°, and their multiples
- Range: Cosecant values are always ≥ 1 or ≤ -1, never between -1 and 1
- Reciprocal Relationship: csc(θ) × sin(θ) = 1 (when both are defined)
Frequently Asked Questions
What is csc(90°)?
csc(90°) = 1. This is because sin(90°) = 1, and csc(θ) = 1/sin(θ), so csc(90°) = 1/1 = 1.
Why is cosecant undefined at 0° and 180°?
Cosecant is undefined when sin(θ) = 0. At 0° and 180° (and 360°, etc.), sin(θ) = 0, so we would be dividing by zero: csc(θ) = 1/0, which is undefined.
What's the relationship between cosecant and sine?
Cosecant is the reciprocal of sine: csc(θ) = 1/sin(θ). When sine is large, cosecant is small, and vice versa. They multiply to 1: sin(θ) × csc(θ) = 1 (when both are defined).
Can cosecant be between -1 and 1?
No! Cosecant values are always ≥ 1 or ≤ -1, never between -1 and 1. This is because sine values are between -1 and 1, so their reciprocals (cosecant) must be outside that range.
How is cosecant used in practice?
Cosecant appears in various applications including wave equations, signal processing, electrical engineering (AC circuits), and in solving certain types of differential equations.
What are the cofunction relationships for cosecant?
csc(θ) = sec(90° - θ). Cosecant and secant are cofunctions - the cosecant of an angle equals the secant of its complement.