Reciprocal Function Domain And Range
Cosecant
The cosecant function is the reciprocal of the trigonometric office sine. Cosecant is one of the main six trigonometric functions and is abbreviated as csc x or cosec x, where x is the angle. In a right-angled triangle, cosecant is equal to the ratio of the hypotenuse and perpendicular. Since it is the reciprocal of sine, we write it as csc x = i / sin 10.
In this article, nosotros will explore the concept of cosecant function and empathise its formula. We will plot the cosecant graph using its domain and range, explore the trigonometric identities of cosec x, its values, and properties. We will solve a few examples based on the concept of csc x to sympathize its applications better.
| 1. | What Is a Cosecant Function? |
| two. | Cosecant Function Formula |
| three. | Domain and Range of Cosec x |
| 4. | Cosecant Graph |
| 5. | Cosecant Identities |
| 6. | Properties of Cosecant Function |
| 7. | Cosecant Values |
| 8. | FAQs on Cosecant Function |
What Is a Cosecant Function?
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 ten, it is defined as the ratio of the length of the hypotenuse and the length of the perpendicular of a correct-angled triangle.
Consider a unit circle with points O equally the center, P on the circumference, and Q inside the circle and join them every bit shown above. Since it is a unit of measurement circumvolve, the length of OP is equal to the 1 unit. Consider the measure of bending POQ equal to x degrees. So, using the cosecant definition, we have
csc ten = OP/PQ
= 1/PQ
Cosecant Office Formula
Since the cosecant function is the reciprocal of the sine function, nosotros can write its formula every bit
Cosec x = i / sin x
Likewise, since the formula for sin 10 is written as
Sin 10 = Perpendicular / Hypotenuse and csc 10 is the reciprocal of sin x, nosotros can write the formula for the cosecant role as
Cosec ten = Hypotenuse / Perpendicular
Domain and Range of Cosec x
As we discussed earlier, cosecant is the reciprocal of the sine office, that is, csc x = 1 / sin x, cosec 10 is divers for all real numbers except for values where sin x is equal to zip. We know that sin x is equal to for all integral multiples of pi, that is, sin x = 0 implies that that x = nπ, where north is an integer. Then, cosec ten is defined for all existent numbers except nπ. At present, nosotros know that the range of sin x is [-1, ane] and csc ten is the reciprocal of sin x, so the range of csc 10 is all existent numbers except (-1, 1). So the domain and range of cosecant are given by,
- 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. Every bit nosotros know cosec x is defined for all existent numbers except for values where sin x is equal to zero. So, we have vertical asymptotes at points where csc x is non defined. Besides, using the values of sin x, we have y = csc 10 as
- When x = 0, sin 10 = 0 and hence, csc x = not divers
- When x = π/6, sin x = ½, csc x = two
- When x = π/four, sin 10 = one/√ii, csc x = √2
- When 10 = π/3, sin 10 = √three/2, csc x = ii/√three
- When x = π/ii, sin x = one, csc x = ane
So, by plotting the above points on a graph and joining them, nosotros take the cosecant graph as follows:
Cosecant Identities
Let us now become through some of the important trigonometric identities of the cosecant function. We utilise these identities to simplify and solve various trigonometric problems.
- 1 + cot²10 = csc²x
- csc (π - x) = csc x
- csc (π/2 - x) = sec 10
- csc (-x) = csc x
- csc 10 = ane / sin x
- csc ten = sec (π/2 - ten)
Properties of Cosecant Function
We have understood that the cosecant function is the reciprocal of the sine function and its formula. Allow u.s.a. now explore some of the important properties of the cosecant function to understand it better.
- The graph of cosec x is symmetrical about the ten-centrality.
- Cosecant Function is an odd function, that is, csc (-10) = -csc x
- The cosecant graph has no x-intercepts, that is, the graph of cosecant does not intersect the 10-axis at any point.
- The value of csc x is positive when sin 10 is positive and it is negative when sin x is negative.
- The menses of csc 10 is 2π radians (360 degrees).
- Cosec x is non defined at the integral multiples of π.
Cosecant Values
To solve various trigonometric problems, we utilise the trigonometry tabular array to memorize the values of the trigonometric functions which are well-nigh commonly used. The tabular array given below shows the values of the cosecant function which help to simplify the problems and are easy to understand and think.
| X (radians) | Csc 10 |
|---|---|
| 0 | Not divers |
| π/6 | 2 |
| π/4 | √2 |
| π/3 | two/√iii |
| π/two | one |
| 3π/2 | -ane |
| 2π | Not defined |
Important Notes on Cosecant Function
- Cosecant is the reciprocal of the sine office.
- 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 Function Examples
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Cosecant Practice Questions
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FAQs on Cosecant
What is Cosecant Function in Trigonometry?
The cosecant function is one of the important six trigonometric functions. Information technology 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 office formula can exist written in two different ways:
- csc ten = i/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 angle in a correct-angled triangle. Nosotros can besides find the cosecant of angle using trigonometric identities.
What is the Difference between Secant and Cosecant?
Secant office 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 adjacent side whereas cosecant is the ratio of the Hypotenuse and Opposite Side.
Is Csc the Inverse of Sin?
No, csc x is non the inverse of sin. Information technology is the reciprocal of the sine function. The inverse of sin is called inverse sine or arcsin.
What is the Reciprocal of Cosecant?
The reciprocal of the cosecant role is the sine function. It is written as sin x = ane/csc x
What is the Period of Cosecant?
The values of the cosecant function repeat later on every 2π radians, so the catamenia of cosec x 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, so cosecant is the reciprocal of sine. As well, the production of these ii functions at an angle is always equal to i. Hence, cosecant is the reciprocal of the sine function.
Is Cosecant Office Graph Continuous?
Cosecant Graph is not continuous as it has vertical asymptotes at points where cosecant function is not defined. We know that cosec 10 is not defined at integer multiples of pi, so the cosecant office graph has a aperture at points nπ, where n is an integer.
Reciprocal Function Domain And Range,
Source: https://www.cuemath.com/trigonometry/cosecant-functions/
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