Cosecant Calculator – Calculate csc(x)

Find the cosecant of an angle using the csc calculator below. Start by entering the angle in degrees or radians.

How to Find the Cosecant of an Angle

In a right triangle, the cosecant of angle α, or csc(α), is the ratio between the angle’s opposite side and the hypotenuse.

illustration of a triangle showing the formula for cosecant being equal to hypotenuse c divided by side a

Cosecant is a trigonometric function abbreviated csc. Use the formula below to calculate the cosecant of an angle.

Cosecant Formula

The cosecant formula is:

csc(α) = hypotenuse copposite a

Thus, the cosecant of angle α in a right triangle is equal to the length of the hypotenuse c divided by the opposite side a.

To solve csc, simply enter the length of the hypotenuse and opposite side, then solve.

This formula might look very similar to the formula to calculate sine. That’s because cosecant is the reciprocal of sine.

Cosecant should not be confused with arcsin, which is the inverse of the sine function. The difference being that cosecant is equal to 1sin(x), while arcsin is the inverse of the sine function.

csc(x) = 1sin(x) = sin(x)-1
arcsin(y) x where y = sin(x)

For example, let’s calculate the cosecant of angle α in a triangle with the length of the hypotenuse equal to 6 and the opposite side equal to 4.

csc(α) = 64
csc(α) = 32

Cosecant Graph

If you graph the cosecant function for every possible angle, it forms a series of repeating U-curves.

graph of the repeating curves representing possible cosecant values

One important property to note in the graph above is that the cosecant of an angle is never in the range of -1 to 1; it’s always smaller than or equal to -1 or larger than or equal to 1.

You’ll also notice that the curves never cross the x-axis at an even multiple of π radians, or 180°

Cosecant Table

The table below shows common angles and the csc value for each of them.

Table showing common angles and csc values for each.
Angle (degrees) Angle (radians) Cosecant
0 undefined
15° π12 √6 + √2
30° π6 2
45° π4 √2
60° π3 2√33
75° 12 √6 – √2
90° π2 1
105° 12 √6 – √2
120° 3 2√33
135° 4 √2
150° 6 2
165° 11π12 √6 + √2
180° π undefined
195° 13π12 -√6 – √2
210° 6 -2
225° 4 -√2
240° 3 2√33
255° 17π12 √2 – √6
270° 2 -1
285° 19π12 √2 – √6
300° 3 2√33
315° 4 -√2
330° 11π6 -2
345° 23π12 -√6 – √2
360° undefined

You might also be interested in our secant and cotangent calculators.