Cosecant Calculator

Welcome to the Cosecant Calculator. This calculator can be used to calculate the cosecant of any angle. The angle can be expressed in degrees or radians.

In a right triangle, the ratio of the hypotenuse to the opposite side of an acute angle is called the cosecant. Cosecant is also one of the trigonometric functions. The cosecant abbreviation is csc.

The cosecant formula is

csc(θ) = hypotenuseopposite

Review another commonly used trigonometric function: sine. The sine is defined as the ratio of the opposite side of the acute angle to the hypotenuse in a right triangle. Just the opposite of cosecant. Therefore, the relationship between cosecant and sine is reciprocal.

As with other trigonometric functions, there are two methods.

When the side lengths are known.

This case is relatively simple. Substitute the relevant values into the cosecant formula. It should be noted that a right triangle needs to give at least 2 side lengths. If it is not the hypotenuse and the opposite side, you need to use the Pythagorean theorem to calculate the other side.

For example, the length of the hypotenuse of a right triangle is 5, and the length of the side adjacent to angle α is 3. What is the cosecant of angle α?

First, find the opposite side.

opposite² + adjacent² = hypotenuse²

opposite² + 3² = 5²

opposite² + 9 = 25

opposite² = 16

opposite = 4

Second, substitute the opposite and hypotenuse into the formula.

csc(α) = hypotenuseopposite = 54

So, the cosecant of angle α is 54.

When the angle degree is known.

In this case, the calculation is usually done with the help of a cosecant calculator. Because the cosecant of degrees is difficult to calculate directly by hand. That’s why we developed this cosecant calculator!

If you don’t have a cosecant calculator, you can use the cosecant table, and find the corresponding cosecant according to degrees or radians.

Now, we will combine the cosecant curve to summarize the properties of the cosecant.

1. Domain – The domain of the cosecant function is all values except kπ (k is an integer).
2. Range – The cosecant range is the same as the secant, the absolute value of the cosecant is greater than or equal to 1.
3. Period – The smallest period of cosecant is 2π. csc(θ) = csc(θ + 2π)
4. Odd function – Since csc(θ) = -csc(-θ), cosecant is an odd function.

Furthermore, in different quadrants of the coordinate axis, the cosecant ranges are also different. Cosecant comparisons for the four quadrants are listed below.

1. Cosecant derivative

The derivative of cosecant is equal to negative cotangent times cosecant. Its derivation process is as follows

(csc(θ))’

= (1sin(θ))’

= -sin(θ)’sin²(θ)

= -cos(θ)sin²(θ)

= -cot(θ) * 1sin(θ)

= -cot(θ) * csc(θ)

2. Inverse cosecant

The inverse of cosecant is arccosecant, denoted as arccsc, or csc^-1. It is a type of inverse trigonometric function.

csc(90°) = 1

csc^-1(1) = arccsc (1) = 90°

3. Reciprocal cosecant

The reciprocal of cosecant is sine, which is equal to the ratio of its opposite side to its hypotenuse side. The abbreviation is sin.

1csc(θ) = sin(θ)

The cosecant calculator is very easy to use, three steps:

First, enter degrees or radians. Such as, 45, π, 2π/5 and so on. If you don’t know how to enter π, just type pi instead.

Second, choose the type, degrees or radians.

Finally, click Calculate button to get the cosecant answer, or click Reset button to start a new calculation.