Cosecant Calculator – Guide & Formulas
Calculate the cosecant (csc) of any angle instantly. Enter an angle in degrees or radians to find csc(x) with step-by-step results.
The Cosecant Calculator computes the cosecant of any angle. Enter an angle in degrees or radians and instantly receive csc(x) = 1/sin(x) with the sine value, step-by-step breakdown, and a reference table of common cosecant values.
Key Takeaway
Use the free Cosecant Calculator to calculate the cosecant (csc) of any angle instantly. enter an angle in degrees or radians to find csc(x) with step-by-step results. Get instant results with step-by-step explanations.
How to Use the Cosecant Calculator
- Enter an angle value in the input field.
- Select degrees or radians as the angle unit.
- The calculator instantly computes the cosecant result.
- Review the sine value, cosecant result, and step-by-step explanation.
The Formula
Variable Definitions
- csc(θ): The cosecant of angle θ: the reciprocal of sine
- θ: The input angle in degrees or radians
- sin(θ): The sine of angle θ: opposite/hypotenuse
Example: Cosecant of 30°
Find csc(30°).
- Step 1: First compute sin(30°) = 0.5.
- Step 2: Apply the formula: csc(30°) = 1 / sin(30°) = 1 / 0.5.
- Step 3: Therefore csc(30°) = 2.
- Step 4: Verify: sin(30°) × csc(30°) = 0.5 × 2 = 1 ✓
Frequently Asked Questions
What is cosecant?
Cosecant (csc) is the reciprocal of the sine function. For any angle θ, csc(θ) = 1/sin(θ). In a right triangle, it is the ratio of the hypotenuse to the opposite side.
What is the domain of cosecant?
The domain of csc(θ) is all real numbers except where sin(θ) = 0, which occurs at θ = nπ (0°, 180°, 360°, etc.). At these points, csc(θ) is undefined.
What is the range of cosecant?
The range of csc(θ) is (-∞, -1] ∪ [1, +∞). The cosecant function never takes values between -1 and 1 exclusive.
What is csc(30°)?
csc(30°) = 1/sin(30°) = 1/0.5 = 2. This is one of the most commonly used cosecant values.
What is csc(90°)?
csc(90°) = 1/sin(90°) = 1/1 = 1. Since sin(90°) = 1, the cosecant at 90° is also 1.
What is csc(0°)?
csc(0°) is undefined because sin(0°) = 0, and dividing by zero is undefined. The cosecant function has a vertical asymptote at 0°.
How is cosecant related to sine?
Cosecant is the reciprocal of sine: csc(θ) = 1/sin(θ). Equivalently, sin(θ) = 1/csc(θ). They are inverse multiplicative pairs.
What is the cofunction of cosecant?
The cofunction of cosecant is secant (sec). The identity is csc(θ) = sec(90° - θ). For example, csc(30°) = sec(60°) = 2.
How do I graph cosecant?
The graph of csc(θ) has vertical asymptotes wherever sin(θ) = 0 (at 0°, 180°, 360°, etc.). Between consecutive asymptotes, the graph forms U-shapes above and below the x-axis.
What are common cosecant values?
Common values: csc(30°) = 2, csc(45°) = √2 ≈ 1.414, csc(60°) = 2/√3 ≈ 1.155, csc(90°) = 1. Undefined at 0°, 180°, 360°.