Cosecant
Reciprocal of sine.
Formula
Variables
Example
csc 30° = 2
How it works
Cosecant, written as csc θ, is defined as the reciprocal of sine, so it tells you "one divided by sine" of an angle. This works because for any nonzero value, taking the reciprocal flips a ratio: if sin θ = opposite/hypotenuse in a right triangle, then csc θ = hypotenuse/opposite. In the unit circle, it is also the reciprocal of the y-coordinate, which is why it becomes undefined whenever sin θ = 0.
When to use it
Use cosecant when a problem gives you sine and you need its reciprocal, or when a formula is written more naturally in terms of csc instead of sin. It is common in trigonometric identities, calculus, and physics problems involving angles and wave behavior.
Worked example, step by step
Find csc 45°. Step 1: Start with the definition: csc θ = 1 / sin θ. Step 2: Substitute θ = 45°: csc 45° = 1 / sin 45°. Step 3: Use the exact value sin 45° = √2 / 2. Step 4: Compute the reciprocal: csc 45° = 1 / (√2 / 2) = 2 / √2. Step 5: Simplify: 2 / √2 = √2. Final answer: csc 45° = √2.
Common mistakes to avoid
- Forgetting that cosecant is the reciprocal of sine, not cosine
- Plugging the angle into degrees/radians incorrectly and using the wrong sine value
- Leaving the answer as a fraction with a denominator of zero when sin θ = 0, even though csc is undefined there
Did You Know?
Cosecant, the reciprocal of sine, shoots off to infinity at 0° — one reason it appears less often in practice.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.
Tangent Identity
BasicTangent equals sine divided by cosine.