Secant (sec), Cosecant (csc), and Cotangent (cot)

This section will show you how the functions of trigonometry like cotangent, secant, and cosecant are related to the other trigonometric functions of sine, cosine, and tangent.

Trigonometric functions

Functions of trigonometric, angles’ functions, are common in the real world and in mathematics. The sound which comes out of the speakers of computers is generated by the waves of trigonometric, or the sound waves transmitted out of the speakers, which are seen in the sine waveform.

At this point, you are already aware of the sine, cosine and tangent functions. These are just three basic functions of trigonometry. All other trigonometric functions are totally based on these three functions, which you will see.

Do you know the functions of sine, cosine, and tangent? If not, then recall it with their three sides of opposite, hypotenuse and adjacent sides. Look at the right-angle triangle, do you remember how we defined those three functions?

The sine function is defined as the opposite side/hypotenuse and the cosine is defined as the adjacent side/ hypotenuse, and the tangent is the ratio of the opposite/adjacent side. Now we have reviewed these three functions of trigonometric, let us see the other three functions of trigonometry of cotangent, secant, and cosecant.

Cotangent

First of all, we have the function of cotangent, which is defined as the reciprocal of the inverse of the function of a tangent. In mathematics, we write it as cot θ = 1 / tan θ. All our trigonometric functions are shortened to three letters when written the function while evaluating.

Now, because the cotangent is the reciprocal of the tangent function, we also define it as the inverse function of the tangent. If the tangent is the opposite/adjacent side, then the cotangent function is the reciprocal of it – adjacent/opposite. We can write all such information like this:

Cotangent function is evaluated as:

Cot θ = 1 / tan θ = adjacent side / opposite side

Secant function

Now we have the function of secant. We can define it as the reciprocal of the cosine function. The three letters, or say a short form of secant is sec. Do you know how cosine is defined? Yes, the cosine is written as – adjacent side/hypotenuse. Then the secant is the reciprocal of the cosine function which is the reciprocal of the cosine function –

Sec θ = 1 / cos θ = hypotenuse / adjacent side.

Cosecant function

Now we have the function of the cosecant. This function is the reciprocal of the sine function which has the short form cosec or CSC. As this is the reciprocal of the sine function and the sine function is defined as the opposite side/hypotenuse, we can alternately define cosecant function as:

Cosecant θ = 1 / sin θ = hypotenuse / opposite side

Thus, here you have a clear idea of the concept of all the three trigonometric functions. Examine the following diagram:

secant

Example 1: What is the csc of x?

A is the opposite side, B is the adjacent side, and C is the hypotenuse.

Since csc = hypotenuse /opposite,

csc x = C/A

Example 2: What is the sec of x?

Again, A is the opposite side, B is the adjacent side, and C is the hypotenuse.

Since sec = hypotenuse / adjacent,

sec x = C/B

Example 3: What is the cot of x?

Since cot = adjacent / opposite,

cot x = B/A