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Cosecant

cosecant
csc
cosec

The trig function cosecant, written csc θ. csc θ equals The fraction 1 over sin θ. For acute angles, csc θ can be found by the SOHCAHTOA definition, shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = csc x is a periodic function with period 2π.

Two diagrams defining csc θ: a right triangle with hypotenuse, opposite, adjacent sides; a unit circle with point (x,y) and...

 

See also

Inverse cosecant, angle

Key Formula

csc⁡θ=1sin⁡θ=hypotenuseopposite\csc\theta = \frac{1}{\sin\theta} = \frac{\text{hypotenuse}}{\text{opposite}}
Where:
  • θ\theta = The angle in a right triangle or on the unit circle
  • hypotenuse\text{hypotenuse} = The longest side of a right triangle, opposite the right angle
  • opposite\text{opposite} = The side of the right triangle across from the angle θ

Worked Example

Problem: In a right triangle, the side opposite angle θ is 5 and the hypotenuse is 13. Find csc θ.
Step 1: Recall the right-triangle definition of cosecant.
csc⁡θ=hypotenuseopposite\csc\theta = \frac{\text{hypotenuse}}{\text{opposite}}
Step 2: Substitute the given values into the formula.
csc⁡θ=135\csc\theta = \frac{13}{5}
Step 3: Simplify or convert to a decimal if needed.
csc⁡θ=2.6\csc\theta = 2.6
Answer: csc θ = 13/5 = 2.6

Another Example

This example uses an angle beyond the first quadrant, showing how to apply reference angles and determine the sign of cosecant using quadrant rules, rather than reading values directly from a right triangle.

Problem: Find the exact value of csc 210°.
Step 1: Identify the reference angle. 210° is in the third quadrant, so the reference angle is 210° − 180° = 30°.
Reference angle=30°\text{Reference angle} = 30°
Step 2: Recall that sin 30° = 1/2. In the third quadrant, sine is negative, so sin 210° = −1/2.
sin⁡210°=−12\sin 210° = -\frac{1}{2}
Step 3: Take the reciprocal to find cosecant.
csc⁡210°=1sin⁡210°=1−12\csc 210° = \frac{1}{\sin 210°} = \frac{1}{-\tfrac{1}{2}}
Step 4: Simplify the result.
csc⁡210°=−2\csc 210° = -2
Answer: csc 210° = −2

Frequently Asked Questions

What is the difference between cosecant and secant?
Cosecant is the reciprocal of sine: csc θ = 1/sin θ. Secant is the reciprocal of cosine: sec θ = 1/cos θ. They correspond to different ratios in a right triangle — cosecant involves the opposite side, while secant involves the adjacent side.
When is cosecant undefined?
Cosecant is undefined whenever sin θ = 0, because you cannot divide by zero. This happens at θ = 0°, 180°, 360°, and more generally at every integer multiple of π (i.e., θ = nπ where n is any integer). On the graph of y = csc x, these points correspond to vertical asymptotes.
Is cosecant the same as the inverse sine function?
No. Cosecant is the reciprocal of sine (csc θ = 1/sin θ), meaning you divide 1 by the sine value. The inverse sine function, written sin⁻¹ x or arcsin x, takes a ratio as input and returns an angle. These are completely different operations.

Cosecant (csc) vs. Sine (sin)

Cosecant (csc)Sine (sin)
DefinitionReciprocal of sine: 1/sin θRatio of opposite to hypotenuse
Formula (right triangle)hypotenuse / oppositeopposite / hypotenuse
Range(−∞, −1] ∪ [1, ∞)[−1, 1]
Undefined atθ = nπ (where sin θ = 0)Never undefined
Period2π2π
When to useWhen you know the sine and need its reciprocal, or when a problem involves hyp/oppWhen you need the basic ratio opp/hyp

Why It Matters

Cosecant appears throughout trigonometry and precalculus courses, especially when simplifying trig identities and solving trig equations. You will encounter it in calculus when working with integrals and derivatives involving reciprocal trig functions. It also shows up in physics and engineering when modeling wave behavior and analyzing forces in non-standard orientations.

Common Mistakes

Mistake: Confusing cosecant with inverse sine (arcsin). Students write csc θ when they mean sin⁻¹(θ), or vice versa.
Correction: Remember: csc θ = 1/sin θ (a reciprocal — you flip the fraction), while sin⁻¹(x) = arcsin(x) (an inverse function — you find the angle). These perform fundamentally different operations.
Mistake: Mixing up cosecant and secant by pairing csc with cosine instead of sine.
Correction: Use this memory aid: cosecant and sine both start with different letters, and that's the pair — csc goes with sin. Similarly, secant (sec) goes with cosine (cos). The 'co-' prefixes are swapped on purpose.

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