what is domain of sec(inverse)
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Answer:
Domain and Range of Sec Inverse x
Hence the domain of sec inverse x is (-∞, -1] U [1, ∞) (being the range of secant function) and the range of arcsec is [0, π/2) U (π/2, π] (principal branch of sec x).
Answer:
Sec inverse x is an important inverse trigonometric function. Sec inverse x is also referred to by different names such as arcsec, inverse secant, and inverse sec x. The range of the trigonometric function sec x becomes the domain of sec inverse x, that is, (-∞, -1] U [1, ∞) and the range of arcsec function is [0, π/2) U (π/2, π]. Please note that sec inverse x is not the reciprocal of the trigonometric function secant x.
Let us study the concept of the inverse trigonometric function sec inverse x, derivative and integral of sec inverse x and understand the properties of the inverse sec with the help of its graph.
What is Sec Inverse x?
Sec Inverse x is the inverse trigonometric function of the secant function. Mathematically, it is denoted by sec-1x. It can also be written as arcsec x. In a right-angled triangle, the secant function is given by the ratio of the hypotenuse and the base, that is, sec θ = Hypotenuse/Base = x (say). Using this, sec inverse x formula is given by θ = sec-1x = sec-1(Hypotenuse/Base). The domain of sec inverse x is given by (-∞, -1] U [1, ∞) and its range is [0, π/2) U (π/2, π]. If sec x = y ⇒ x = sec-1y. Let us see a few examples of how to determine the values of sec inverse x.
If sec 0 = 1 ⇒ sec-1(1) = 0
If sec (π/6) = 2/√3 ⇒ sec-1(2/√3) = π/6
If sec (π/4) = √2 ⇒ sec-1(√2) = π/4
If sec (π/3) = 2 ⇒ sec-1(2) = π/3
Domain and Range of Sec Inverse x
We know that sec inverse x is an inverse function of the secant function. Also, it is necessary for a function to be bijective (one-one and onto) to have an inverse. Hence, to make the secant function bijective, we take the principal value branch as the range of sec inverse x. Hence the domain of sec inverse x is (-∞, -1] U [1, ∞) (being the range of secant function) and the range of arcsec is [0, π/2) U (π/2, π] (principal branch of sec x).
Sec Inverse x Graph
Now, let us understand more about sec inverse x and check its domain and range using the graph of arcsec. We can plot the graph of sec inverse x using some of its points. By plotting the below-given points on a graph with the angle on Y-axis and real numbers on X-axis given by y = sec-1x:
If x = 1, y = 0
If x = 2/√3, y = π/6
If x = √2, y = π/4
If x = 2, y = π/3
Sec Inverse x Graph
Sec Inverse x Derivative and Integration
Now, we will determine the derivative of sec inverse x and the integral of sec inverse x. To determine the derivative of arcsec, we will use implicit differentiation. Assume y = sec-1x ⇒ sec y = x. We need to determine the value of dy/dx. Now, differentiate sec y = x with respect to x. We have,
d(sec y)/dy × dy/dx = 1
⇒ sec y tan y × dy/dx = 1
⇒ dy/dx = 1/sec y tan y [Because d(sec θ)/dθ = sec θ tan θ]
⇒ dy/dx = 1/sec y √(sec2y - 1) [Using 1 + tan2θ = sec2θ ⇒ tan θ = √(sec2θ - 1)]
⇒ dy/dx = 1/x √(x2 - 1) [Because sec y = x]
Hence sec inverse x derivative is 1/x √(x2 - 1)
Similarly, we can determine the integral of sec inverse x using integration by parts. The sec inverse x integral is given by ∫sec-1x dx = x sec-1x - ln |x + √(x2 - 1)| + C, where C is the constant of integration.
Sec Inverse x Formulas
Now, that we have explored the concept of sec inverse x, let us now see some formulas of sec inverse x that are used to solve various mathematical trigonometric problems:
sec-1(-x) = π - sec-1x
sec-1x + cosec-1x = π/2
cos-1x = sec-1(1/x)
cos-1(1/x) = sec-1x
Important Notes on Sec Inverse x
Sec Inverse x derivative: d(sec-1x)/dx = 1/x √(x2 - 1)
Arcsec Integral: ∫sec-1x dx = x sec-1x - ln |x + √(x2 - 1)| + C
Sec Inverse x is an inverse trigonometric function of secant function
Domain of Arcsec: (-∞, -1] U [1, ∞)
Range of Sec inverse x: [0, π/2) U (π/2, π]
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Sec Inverse x Examples
Example 1: Determine the value of sec-1(-2/√3) using sec inverse x formulas
Solution: We know that sec-1(2/√3) = π/6 and sec-1(-x) = π - sec-1x. Therefore, we have
sec-1(-2/√3) = π - sec-1(2/√3)
⇒ sec-1(-2/√3) = π - π/6
⇒ sec-1(-2/√3) = 5π/6
Answer: The value of sec-1(-2/√3) is 5π/6.
Example 2: If cos-1(1/x) is 0.56, then what is the value of arcsec x.
Solution: Using sec inverse x formula, we have cos-1(1/x) = sec-1x, therefore sec-1x = 0.56
Answer: arcsec x = 0.56
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Sec Inverse x Problems
Q. 1 What is derivative of sec inverse x?
1/√(x2 + 1)
1/√(x, 2, + 1)
1/√(x2 - 1)
1/√(x, 2, - 1)
1/x √(x2 + 1)
1/x √(x, 2, + 1)
1/x √(x2 - 1)
1/x √(x, 2, - 1)
Q. 2 Identify the Range of Sec Inverse x?
[0, π/2]
[0, π/2]
[0, π]
[0, π]
[0, π] - {π}
[0, π] - {π}
[0, π] - {π/2}
[0, π] - {π/2}