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.
| 1. | What is Csc Sec Cot? |
| 2. | Csc Sec Cot Formula |
| 3. | Domain and Range of Cosecant, Secant, and Cotangent Functions |
| 4. | Csc Sec Cot Graph |
| 5. | Csc Sec Cot Chart |
| 6. | Csc Sec Cot Identities |
| 7. | FAQs on 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:
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 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.
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.
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:
We can also write csc sec cot formulas in terms of sin cos tan as given below:
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 | (-∞, +∞) |
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.
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 |
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:
The above identities help us to solve the trigonometric problems with easy calculations.
Important Notes on Csc Sec Cot
☛ Related Topics:
Example 1: Find the value of csc x, if the hypotenuse = 5 units and the side adjacent to x is 4 units in a right-angled triangle. Solution: We know that csc x = Hypotenuse / Opposite Side. Using Pythagoras theorem, we have Hypotenuse 2 = Adjacent Side 2 + Opposite Side 2 ⇒ Opposite Side 2 = Hypotenuse 2 - Adjacent Side 2 = 5 2 - 4 2 = 25 - 16 = 9 Opposite Side = 3 units So, csc x = 5/3 Answer: csc x = 5/3
Example 2: Find the value of cot x if csc x = 13/12 and sec x = 13/5. Solution: We know that csc x = 1/sin x and sec x = 1/cos x ⇒ 1/sin x = 13/12 and 1/cos x = 13/5 ⇒ sin x = 12/13 and cos x = 5/13 So, cot x = cos x / sin x = (5/13) / (12/13) = 5/12 Answer: cot x = 5/12
Example 3: Calculate the value of sec x if cos x = 7/25 Solution: We know that sec x = 1/cos x. So, we have sec x = 1 / cos x = 1 / (7/25) = 25/7 Answer: sec x = 25/7
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