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Cosecant: Definition, Function & Formula

Instructor: Ellen Manchester
In this lesson, we will be learning about one of the six trigonometric functions. Cosecant is a function that has a very close relationship with the sine function. Let's see what their relationship is all about.

Cosecant: What Is It?

Trigonometric functions are useful in many areas of our world. Many careers use these functions to work with the relationships of distances based on a right triangle, this is where the term triangulation comes from. Criminal investigations, forensics, and video games are just three areas where trigonometric functions are useful.

In this lesson, we will be learning about one of these trigonometric functions, cosecant, abbreviated as csc. As with all trigonometric functions, cosecant is based on a right triangle. The cosecant function is the reciprocal of the sine function. The sine function is the opposite side divided by the hypotenuse, so the cosecant function is the hypotenuse divided by the opposite side.

The side opposite the right angle is called the hypotenuse, and the other two sides are called legs. Which angle we are using will determine whether the sides are opposite or adjacent. Take a look at these two right triangles. As you can see, the names of the legs are determined by which angle we are working with.

Trig triangle

trigtriangle2

Hyp is the abbreviation of hypotenuse. The side Opp is the side opposite the given angle, and Adj is the side adjacent or next to the given angle. These functions give a relationship or ratio between the sides based on the given angles.

The Sine Function

Let's review the sine function first before we discuss the cosecant any more.

The sine function is a ratio based on the opposite side to the hypotenuse.

sine function

To find the sine of a 45-degree angle, we can use our 45-45-90 triangle.

45-45-90 triangle
45-45-90triangle

sine of 45-degree

Remember, we cannot have a square root in the denominator, so we multiply both top and bottom by square root of 2, which eliminates the square root in the denominator. This is called rationalizing the denominator.

Cosecant: Flippin' Sine

The cosecant is the reciprocal, or flipped version, of the sine function:

cosecant

Using the 45-45-90 triangle, we can see it is the reciprocal of our sine function:

csc45

Here is the 30-60-90 triangle.

30-60-90 triangle
30-60-90 triangle

Finding the cosecant of the 30-degree angle, we see the hypotenuse is 2 and the side opposite the 30-degree angle is 1, so csc 30 = 2/1 = 2.

The cosecant of the 60-degree angle is:

csc60

We get this since we cannot have the radical in the denominator and we rationalized the denominator by multiplying both top and bottom by square root of 3.

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