Is Csc Odd Or Even
Cosecant
The cosecant function is the reciprocal of the trigonometric part sine. Cosecant is one of the main six trigonometric functions and is abbreviated equally csc ten or cosec x, where x is the angle. In a correct-angled triangle, cosecant is equal to the ratio of the hypotenuse and perpendicular. Since information technology is the reciprocal of sine, we write information technology as csc x = ane / sin x.
In this article, nosotros will explore the concept of cosecant function and understand its formula. We will plot the cosecant graph using its domain and range, explore the trigonometric identities of cosec x, its values, and properties. Nosotros volition solve a few examples based on the concept of csc x to understand its applications better.
| one. | What Is a Cosecant Function? |
| 2. | Cosecant Function Formula |
| iii. | Domain and Range of Cosec ten |
| 4. | Cosecant Graph |
| v. | Cosecant Identities |
| 6. | Backdrop of Cosecant Function |
| 7. | Cosecant Values |
| 8. | FAQs on Cosecant Function |
What Is a Cosecant Role?
Cosecant is the reciprocal of sine. We have six important trigonometric functions:
- Sine
- Cosine
- Tangent
- Cotangent
- Secant
- Cosecant
Since it is the reciprocal of sin x, it is defined every bit the ratio of the length of the hypotenuse and the length of the perpendicular of a right-angled triangle.
Consider a unit circle with points O as the center, P on the circumference, and Q inside the circle and bring together them equally shown above. Since it is a unit circle, the length of OP is equal to the 1 unit. Consider the measure of angle POQ equal to x degrees. Then, using the cosecant definition, we accept
csc x = OP/PQ
= 1/PQ
Cosecant Function Formula
Since the cosecant function is the reciprocal of the sine role, nosotros can write its formula equally
Cosec x = one / sin x
As well, since the formula for sin x is written as
Sin x = Perpendicular / Hypotenuse and csc ten is the reciprocal of sin ten, nosotros tin write the formula for the cosecant office as
Cosec x = Hypotenuse / Perpendicular
Domain and Range of Cosec x
As we discussed before, cosecant is the reciprocal of the sine office, that is, csc x = 1 / sin 10, cosec ten is defined for all real numbers except for values where sin 10 is equal to zero. Nosotros know that sin x is equal to for all integral multiples of pi, that is, sin 10 = 0 implies that that x = nπ, where due north is an integer. So, cosec x is defined for all existent numbers except nπ. Now, we know that the range of sin x is [-i, ane] and csc x is the reciprocal of sin x, and then the range of csc x is all real numbers except (-1, 1). And so the domain and range of cosecant are given past,
- Domain = R - nπ
- Range = (-∞, -1] U [+1, +∞)
Cosecant Graph
Now that we know the domain and range of cosecant, let us now plot its graph. As we know cosec ten is divers for all real numbers except for values where sin x is equal to nada. So, we have vertical asymptotes at points where csc x is not defined. As well, using the values of sin x, nosotros have y = csc x as
- When ten = 0, sin x = 0 and hence, csc x = non defined
- When x = π/6, sin x = ½, csc x = two
- When 10 = π/iv, sin ten = ane/√two, csc 10 = √2
- When x = π/3, sin x = √3/ii, csc x = 2/√3
- When x = π/2, sin x = one, csc ten = ane
So, by plotting the to a higher place points on a graph and joining them, we have the cosecant graph equally follows:
Cosecant Identities
Let us now become through some of the important trigonometric identities of the cosecant office. Nosotros use these identities to simplify and solve diverse trigonometric problems.
- 1 + cot²10 = csc²ten
- csc (π - x) = csc x
- csc (π/2 - x) = sec x
- csc (-x) = csc x
- csc x = i / sin x
- csc ten = sec (π/two - x)
Properties of Cosecant Office
We take understood that the cosecant office is the reciprocal of the sine function and its formula. Let us at present explore some of the of import properties of the cosecant function to empathize it meliorate.
- The graph of cosec x is symmetrical nigh the x-centrality.
- Cosecant Function is an odd function, that is, csc (-ten) = -csc x
- The cosecant graph has no ten-intercepts, that is, the graph of cosecant does not intersect the x-axis at whatsoever point.
- The value of csc ten is positive when sin x is positive and it is negative when sin x is negative.
- The period of csc x is 2π radians (360 degrees).
- Cosec x is not divers at the integral multiples of π.
Cosecant Values
To solve various trigonometric problems, we use the trigonometry table to memorize the values of the trigonometric functions which are most commonly used. The table given below shows the values of the cosecant function which help to simplify the problems and are easy to empathize and recollect.
| 10 (radians) | Csc x |
|---|---|
| 0 | Not defined |
| π/half dozen | 2 |
| π/4 | √2 |
| π/iii | 2/√3 |
| π/2 | 1 |
| 3π/2 | -1 |
| 2π | Not divers |
Important Notes on Cosecant Function
- Cosecant is the reciprocal of the sine role.
- It is equal to the ratio of hypotenuse and perpendicular of the right angles triangle.
- The cosecant graph has vertical asymptotes and has no x-intercepts.
- Cosecant Function is defined at integer multiples of π.
☛ Related Topics:
- Cosine
- Trigonometric Tabular array
- Trigonometric Ratios
Cosecant Role Examples
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Cosecant Practice Questions
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FAQs on Cosecant
What is Cosecant Part in Trigonometry?
The cosecant part is one of the important vi trigonometric functions. It is the reciprocal of the sine role and hence, is equal to the ratio of Hypotenuse and Perpendicular of a correct-angled triangle.
What is Cosecant Function Formula?
The cosecant function formula tin can be written in two different ways:
- csc x = one/sin x
- csc x = Hypotenuse/Perpendicular OR Hypotenuse/Opposite Side
What is the Cosecant of an Bending?
The cosecant of an angle is equal to the ratio of the hypotenuse and opposite side of the bending in a right-angled triangle. We can also find the cosecant of bending using trigonometric identities.
What is the Difference between Secant and Cosecant?
Secant part is the reciprocal of the cosine part and the Cosecant function is the reciprocal of the sine function. Secant is the ratio of hypotenuse and side by side side whereas cosecant is the ratio of the Hypotenuse and Opposite Side.
Is Csc the Changed of Sin?
No, csc ten is not the inverse of sin. Information technology is the reciprocal of the sine function. The changed of sin is chosen inverse sine or arcsin.
What is the Reciprocal of Cosecant?
The reciprocal of the cosecant function is the sine function. It is written as sin x = one/csc x
What is the Period of Cosecant?
The values of the cosecant function repeat after every 2π radians, so the menses of cosec 10 is equal to 2π radians (360 degrees).
Why is Cosecant the Reciprocal of Sine?
We know that sin x is the ratio of perpendicular and Hypotenuse of a right-angled triangle and Cosecant is the ratio of perpendicular and Hypotenuse, and then cosecant is the reciprocal of sine. Too, the production of these 2 functions at an angle is always equal to one. Hence, cosecant is the reciprocal of the sine function.
Is Cosecant Function Graph Continuous?
Cosecant Graph is not continuous as it has vertical asymptotes at points where cosecant function is not defined. We know that cosec ten is not defined at integer multiples of pi, then the cosecant function graph has a discontinuity at points nπ, where n is an integer.
Is Csc Odd Or Even,
Source: https://www.cuemath.com/trigonometry/cosecant-functions/
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