Domain And Range of Trigonometric Function
Before discussing trigonometric function domain range, let's brush up on what a domain and range of any function are. A function is nothing but defined as the rules which are applied to the values inputted. The domain of a function f(x) is defined as the set of all possible values of x such that function f(x) is well defined whereas the range of the function f(x) is defined as the set of all possible values the function f(x) can consider, where x is any number from the domain of a function.
In other words, the set of all possible values that can be used as an input for the function is the domain of a function. For example, for the function f(k) = k, the input value can never be negative as the square root of the negative number is not a real number. On the other hand, the range is the set of all output values for the different values that are inputted. For example, for the function f(x) = x² + 2, the range would be { 2,3,4}.
Domain Range Of Trigonometric Function Along With Their Graphs
Mentioned below are the domain and range of all trigonometric functions such as sine, cosine, tan, sec, cosec, and cot along with their graph for better understanding.
Domain And Range For Sine Function
y = f(x) = sin (x)
Domain: The domain of the sine function is determined for all x real values
Range : \[-1\leq y\leq 1\]
Period: 2\[\pi\]= 360º
Here, the sine function is an odd function
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(Image Will be Uploaded Soon)
Domain And Range For Cosine Function
y = f(x) = cos(x)
Domain: The domain of cosine function is determined for all x real values
Range : \[-1\leq y\leq 1\]
Period: 2\[\pi\]= 360º
Here cosine function is an even function
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(Image Will be Uploaded Soon)
(Image Will be Uploaded Soon)
In the above graph, we can observe that the graph reduces but the range remains the same.
Domain And Range of Tangent Function
Here function (y) = tan (x) comprises vertical asymptotes at \[\pm \frac{(2n+1)\pi }{2}\]. Hence, for - y = f(x) = tan (x).
Domain = The domain of tangent function defined for all x values, except x (2n + 1)(n/2), where n is an integer.
Range = All real numbers ( or y R).
Period: 2\[\pi\]
Here, tangent is an odd function.
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Domain And Range For Secant Function
y = f(x) = sec(x)
Domain: The domain of secant function is defined for all real values of x except x ≠(2n + 1)(π/2) where n is any value of the integer
Range : (-∞,-1] ∪ [1,∞)
Period: 2\[\pi\]
Secant is an even function
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Domain And Range For Cosecant Function
y = f(x) = cosec(x)
Domain: The domain of the cosecant function is defined for all x real values; except x ≠n where n is an integer value.
Range :(-∞,-1] ∪ [1,∞)
Period: 2\[\pi\]
Cosecant is an odd function.
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Domain And Range For Cotangent Function
y = f(x) = cot(x)
Domain: The domain of cotangent function for all the x real values; except x ≠nπ, where n is any value of an integer
Range: All the real numbers
Period:\[\pi\]
Cotangent is an odd function
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Domain And Range of Trigonometric Function: Tabular Form
To make the students understand the trigonometry domain and range more clearly, we have constructed a table below which clearly describes the domain and range of trigonometric functions.
Domain And Range of Inverse Trigonometric Function
Following are the domain and range of all inverse trigonometric functions such as arcsin, arccosine, arctan, arcsec, arccosec, and arccot along with their graph for better understanding.
ArcSin
The inverse sine function y = sin⁻¹(x) means x = sin y.
The inverse sine function is often known as the arcsine function and represented as arcsin x.
y = sin⁻¹ has domain [-1, 1] and range \[-\frac{\pi }{2}\], \[-\frac{\pi }{2}\]
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ArcCosine
The inverse cosine function y = cos⁻¹(x) means x = cos y.
The inverse cosine function is often known as the arccosine function and represented as arccos x.
y = cos⁻¹ has domain [-1, 1] and range [0,\[\pi\] .
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ArcTan
The inverse tangent function y = tan⁻¹(x) means x = tan y.
The inverse tangent function is also known as the arctangent function and is written as arctangent x.
ArcTan y = tan⁻¹ has a restricted domain of all real number values and range \[-\frac{\pi }{2}\], \[-\frac{\pi }{2}\]
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ArcSecant
The inverse secant function, written as arcsecant or sec⁻¹ x is a function whose domain is R - (1,1) whereas the range of arcsecant or sec⁻¹ x is [0,\[\pi\] , \[-\frac{\pi }{2}\]
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ArcCosecant
The inverse cosecant function, written as arccosecant or cosec⁻¹ x is a function whose domain is R whereas the range of arccosecant or cosec⁻¹ x is \[\frac{\pi }{2}\], \[\frac{\pi }{2}\].
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ArcCotangent
The inverse cotangent function, written as arccotangent or cot⁻¹ x is a function whose domain is R whereas the range of arccotangent or cot⁻¹ x is {0,\[\pi\] }.
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Domain And Range of Inverse Trigonometric Function
To make the students understand the inverse trigonometry domain and range more clearly, we have constructed a table below which clearly describes the domain and range of inverse trig functions.
Solved Example
1. What is the domain and range of y = cos(x) – 5?
Solution:
Here,
Domain: x ∈ R
Range: - 4 ≤ y ≤ - 3, y ∈ R
Notice that the range is simply shifted down by 5 units.
2. What is the value of Sin⁻¹ (-1/2)?
Solution:
Sin⁻¹ (-1/2) = y
Sin y = -½ = sin(\[-\pi\]\/6)
or y = \[-\pi\]\/6
FAQs on Trigonometric Function Domain Range
1. What do the terms 'domain' and 'range' signify in the context of trigonometric functions?
In trigonometry, the domain of a function is the set of all possible input values (angles, usually in radians or degrees) for which the function is defined. The range is the set of all possible output values that the function can produce. For example, the angle 'x' is the input for sin(x), and the resulting value is the output.
2. What are the domain and range for the primary trigonometric functions: sine, cosine, and tangent?
The domain and range for the primary trigonometric functions as per the NCERT syllabus are:
- Sine (sin x): The domain is all real numbers (ℝ), as you can take the sine of any angle. The range is the closed interval [-1, 1].
- Cosine (cos x): Similar to sine, the domain is all real numbers (ℝ). The range is also the closed interval [-1, 1].
- Tangent (tan x): The domain includes all real numbers except for odd multiples of π/2 (i.e., x ≠ (2n+1)π/2), where the cosine in the denominator is zero. The range is all real numbers (ℝ).
3. What are the domain and range for the reciprocal trigonometric functions: cosecant, secant, and cotangent?
The domain and range for the reciprocal functions are determined by where their base functions (sine, cosine, tangent) are zero:
- Cosecant (csc x = 1/sin x): The domain is all real numbers except integer multiples of π (x ≠ nπ), where sin x is zero. The range is (-∞, -1] ∪ [1, ∞).
- Secant (sec x = 1/cos x): The domain is all real numbers except odd multiples of π/2 (x ≠ (2n+1)π/2), where cos x is zero. The range is also (-∞, -1] ∪ [1, ∞).
- Cotangent (cot x = 1/tan x): The domain is all real numbers except integer multiples of π (x ≠ nπ), where sin x (in the denominator of cot x = cos x / sin x) is zero. The range is all real numbers (ℝ).
4. Why is the range of sine (sin x) and cosine (cos x) functions restricted to the interval [-1, 1]?
The restricted range of sine and cosine is best understood using the unit circle concept from the CBSE syllabus. For any angle on a circle with a radius of 1, the cosine of the angle is the x-coordinate, and the sine is the y-coordinate. Since the coordinates on a unit circle can never be greater than 1 or less than -1, the values of sin x and cos x are naturally confined to the interval [-1, 1].
5. Why do the tangent (tan x) and cotangent (cot x) functions have a range of all real numbers?
Unlike sine and cosine, tangent is a ratio (tan x = sin x / cos x). As the angle x approaches an odd multiple of π/2 (like 90°), cos x approaches 0. Dividing a non-zero number (sin x) by a number very close to zero results in a value that can be infinitely large or infinitely small. This is why the graph of tan(x) has vertical asymptotes and its range covers all real numbers (ℝ). A similar logic applies to cot(x) as it approaches multiples of π.
6. What are the domain and range of the main inverse trigonometric functions?
To be true functions, inverse trigonometric functions have a restricted domain and range:
- sin⁻¹(x): Domain is [-1, 1], Range is [-π/2, π/2].
- cos⁻¹(x): Domain is [-1, 1], Range is [0, π].
- tan⁻¹(x): Domain is all real numbers (ℝ), Range is (-π/2, π/2).
7. What is the importance of restricting the domain of a standard trigonometric function to define its inverse?
Standard trigonometric functions like sin(x) are periodic, meaning they repeat their values and fail the horizontal line test. A function must be one-to-one (pass the horizontal line test) to have an inverse. By restricting the domain (for example, restricting sin(x) to [-π/2, π/2]), we make the function one-to-one within that interval. This allows us to define a unique inverse function, like sin⁻¹(x), which is a crucial concept in Class 12 Maths.
8. How do transformations, like adding or subtracting a constant, affect the range of a trigonometric function?
Transformations can shift or scale the range of a trigonometric function, but they usually do not affect the domain. For example, consider the function y = cos(x) – 5. The standard range of cos(x) is [-1, 1]. Subtracting 5 from every output value shifts the entire range down by 5 units. Therefore, the new range becomes [-1-5, 1-5], which is [-6, -4]. The domain remains all real numbers.
