Csc Sec Cot
Csc Sec Cot is the abbreviated form of writing the trigonometric functions cosecant, secant, and cotangent functions. We have mainly six trigonometric functions - sine, cosine, tangent, cosecant, secant, and cotangent. Csc sec cot are based on the other three trigonometric functions sin, cos, and tan, respectively. Since csc sec cot are the reciprocals of sin, cos, tan, respectively, they are also called the reciprocal trigonometric functions.
In this article, we will explore the concept of csc sec cot and discuss their domain and range, formulas and plot the graphs. We will also understand the trigonometric formulas involving csc sec cot and their values, and solve a few examples based on using cosecant, secant, and cotangent functions for a better understanding.
What is Csc Sec Cot?
Csc sec cot are the three trigonometric functions cosecant, secant, and cotangent respectively. These functions are also called the reciprocal trigonometric functions as they are the reciprocals of the sine function, cosine function, and tangent function, respectively. Let us understand the three csc sec cot functions separately:
Cosecant Function
The cosecant function is the reciprocal of the sine function and vice-versa. It is defined as the ratio of the hypotenuse and opposite side of the angle in a right-angled triangle. Cosecant is one of the main six trigonometric functions and is abbreviated as csc x or cosec x, where x is the angle.
Secant Function
Secant function is defined as the ratio of the hypotenuse and the adjacent side of the angle in a right-angled triangle. It is the reciprocal of the cosine function. Mathematically, it is written as sec x, where x is the angle. Since it is the reciprocal of cos, it is written as sec x = 1 / cos x.
Cotangent Function
The cotangent function is the reciprocal of the tangent function. So, it is defined as the ratio of the base and perpendicular of a right-angled triangle. We denote the cotangent function as cot x, where x is the angle in degrees or radians. Since cot is the reciprocal of tan, we can write it as cot x = 1 / tan x.
Csc Sec Cot Formula
Now that we know the definition of the csc sec cot functions, let us now go through their formulas. We use these formulas to solve various trigonometric problems. The formulas for csc sec cot are given below:
- csc x = Hypotenuse / Perpendicular (OR) Hypotenuse / Opposite Side
- sec x = Hypotenuse / Base (OR) Hypotenuse / Adjacent Side
- cot x = Perpendicular / Base (OR) Opposite Side / Adjacent Side
We can also write csc sec cot formulas in terms of sin cos tan as given below:
- csc x = 1/sin x
- sec x = 1/cos x
- cot x = 1/tan x
Domain and Range of Cosecant, Secant, and Cotangent Functions
Csc x is defined for all real numbers except for values where sin x is equal to zero, that is, nπ, where n is an integer. The range of the cosecant function is all real numbers except (-1, 1). The domain of the sec x is all real numbers except for points where cosine is not defined, that is, (2n + 1)π/2, where n is an integer. The range of secant function is the set of all real numbers with a magnitude greater than or equal to 1. On the other hand, cotangent is not defined for real numbers where the tangent is equal to zero. The tangent function is equal to zero where sine is equal to zero. So, the cot function is defined for all real numbers except nπ. The range of cotangent is all real numbers.
The table given below gives the domain and range of csc sec cot:
| Function | Domain | Range |
|---|---|---|
| Csc | R - nπ | (-∞, -1] U [+1, +∞) |
| Sec | R - (2n + 1)π/2 | (-∞, -1] U [+1, +∞) |
| Cot | R - nπ | (-∞, +∞) |
Csc Sec Cot Graph
Now, we know the domain and range of csc cot sec functions. Let us now plot their graphs, by plotting the points and joining them. The graphs of csc sec cot have vertical asymptotes as they are not defined at certain points.
Csc Sec Cot Chart
To solve various trigonometric problems, we use the trigonometry table. We memorize some of the commonly used values of the trigonometric functions for easy calculations. The table given below shows the values of csc sec cot for specific angles that are commonly used for quick calculations of various trigonometric questions.
| Angle (in radians) | Csc | Sec | Cot |
|---|---|---|---|
| 0 | Not Defined | 1 | Not Defined |
| π/6 | 2 | 2/√3 | √3 |
| π/4 | √2 | √2 | 1 |
| π/3 | 2/√3 | 2 | 1/√3 |
| π/2 | 1 | Not defined | 0 |
| π | Not Defined | -1 | Not Defined |
| 3π/2 | -1 | Not Defined | - |
| 2π | Not Defined | 1 | Not Defined |
| -π/2 | -1 | Not Defined | 0 |
| -π | Not Defined | -1 | Not Defined |
Csc Sec Cot Identities
So far we have understood the concept of cosecant, secant, and cotangent functions. Let us now explore trigonometric formulas and identities that involve csc sec cot. These identities simplify the trigonometric problems and make the calculations easy. Given below is a list of csc sec cot identities:
- 1 + cot2x = csc2x
- 1 + tan2x = sec2x
- cot x = csc x / sec x
- cot x = cos x / sin x
- csc (π/2 - x) = sec x
- sec (π/2 - x) = csc x
- cot (π/2 - x) = tan x
- tan (π/2 - x) = cot x
- cot (π - x) = - cot x
- csc (π - x) = csc x
- sec (π - x) = - sec x
The above identities help us to solve the trigonometric problems with easy calculations.
Important Notes on Csc Sec Cot
- Csc sec cot are abbreviated forms of trigonometric functions cosecant, secant, and cotangent, respectively.
- Csc sec cot are the reciprocals of sin cos tan, respectively.
- The product of csc sec cot is equal to 1/sin2.
☛ Related Topics:
FAQs on Csc Sec Cot
What is Csc Sec Cot in Trigonometry?
Csc sec cot are trigonometric functions named as, cosecant secant and cotangent, respectively. Csc sec cot are the reciprocal functions of sin cos tan, respectively.
What are Csc Sec Cot Formulas?
The formulas of csc sec cot are as follows:
- csc x = Hypotenuse / Perpendicular (OR) Hypotenuse / Opposite Side
- sec x = Hypotenuse / Base (OR) Hypotenuse / Adjacent Side
- cot x = Perpendicular / Base (OR) Opposite Side / Adjacent Side
What is the Domain and Range of Cosecant, Secant, and Cotangent Functions?
The domain and range of csc sec cot is given in the table below:
| Function | Domain | Range |
|---|---|---|
| Csc | R - nπ | (-∞, -1] U [+1, +∞) |
| Sec | R - (2n + 1)π/2 | (-∞, -1] U [+1, +∞) |
| Cot | R - nπ | (-∞, +∞) |
How to Use Csc Sec Cot?
We can use csc sec cot by substituting their formulas into the problem and hence simplify it. The formulas are:
- csc x = Hypotenuse / Perpendicular (OR) Hypotenuse / Opposite Side
- sec x = Hypotenuse / Base (OR) Hypotenuse / Adjacent Side
- cot x = Perpendicular / Base (OR) Opposite Side / Adjacent Side
We can also use their identities to make calculations easy.
What Does Csc Sec Cot Stand For?
Csc sec cot stands for cosecant secant and cotangent functions in trigonometry.
What are Csc Sec Cot Formulas in Terms of Sin Cos Tan?
We can write csc sec cot formulas in terms of sin cos tan as given below:
- csc x = 1/sin x
- sec x = 1/cos x
- cot x = 1/tan x