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Questions and Answers
What is the range of the function sin–1?
What is the range of the function sin–1?
Which of the following functions is defined for the range R – (–1, 1)?
Which of the following functions is defined for the range R – (–1, 1)?
What is the correct interpretation of sin–1x?
What is the correct interpretation of sin–1x?
What is the principal value of cot–1(3)?
What is the principal value of cot–1(3)?
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Which function has a range of (0, π)?
Which function has a range of (0, π)?
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What is the principal value range for the sec–1 function?
What is the principal value range for the sec–1 function?
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For which value is sin–1(1/2) equal to?
For which value is sin–1(1/2) equal to?
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Which of the following statements is false regarding inverse trigonometric functions?
Which of the following statements is false regarding inverse trigonometric functions?
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Which of the following ranges belongs to the function sin–1?
Which of the following ranges belongs to the function sin–1?
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What is the result of sin–1(−1)?
What is the result of sin–1(−1)?
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If sin–1 x = y, which statement is true about y?
If sin–1 x = y, which statement is true about y?
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Which expression evaluates to π/3?
Which expression evaluates to π/3?
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What is the sum of tan–1(1) and cos–1(1/2)?
What is the sum of tan–1(1) and cos–1(1/2)?
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What is the value of tan–1(−3) given the properties of inverse functions?
What is the value of tan–1(−3) given the properties of inverse functions?
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What is the valid range for x when using the equation $y = sin^{-1} x$?
What is the valid range for x when using the equation $y = sin^{-1} x$?
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Which identity is correct for the equation $sin^{-1}(sin y)$?
Which identity is correct for the equation $sin^{-1}(sin y)$?
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What is the result of $sin^{-1}(2sinθcosθ)$?
What is the result of $sin^{-1}(2sinθcosθ)$?
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When working with inverse trigonometric functions, which statement is correct about their domains?
When working with inverse trigonometric functions, which statement is correct about their domains?
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For the relation $sin^{-1}(2x(1-x^2)) = 2sin^{-1} x$, what is the condition for x?
For the relation $sin^{-1}(2x(1-x^2)) = 2sin^{-1} x$, what is the condition for x?
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In the equation $sin^{-1}(2x(1-x^2)) = 2cos^{-1} x$, what is the required range for x?
In the equation $sin^{-1}(2x(1-x^2)) = 2cos^{-1} x$, what is the required range for x?
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What is the expression for $tan^{-1}$ in its simplest form as mentioned?
What is the expression for $tan^{-1}$ in its simplest form as mentioned?
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What is the equivalent relationship when $y = cos^{-1}x$?
What is the equivalent relationship when $y = cos^{-1}x$?
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What is the range of the function y = sin–1 x?
What is the range of the function y = sin–1 x?
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Which of the following equations relates to y = tan–1 x?
Which of the following equations relates to y = tan–1 x?
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What is the domain of the function y = cosec–1 x?
What is the domain of the function y = cosec–1 x?
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For the equation sin–1(1 – x) – 2 sin–1 x = π/2, what is one possible value of x?
For the equation sin–1(1 – x) – 2 sin–1 x = π/2, what is one possible value of x?
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Which statement correctly describes the inverse trigonometric function y = cos–1 x?
Which statement correctly describes the inverse trigonometric function y = cos–1 x?
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What is the relationship between sin(tan–1 x) and x for |x| < 1?
What is the relationship between sin(tan–1 x) and x for |x| < 1?
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Which of the following accurately describes the output range of y = sec–1 x?
Which of the following accurately describes the output range of y = sec–1 x?
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What is the simplest form of cot(2/(x - 1)), where x > 1?
What is the simplest form of cot(2/(x - 1)), where x > 1?
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Which identity is used when rewriting tan^(-1)(cos x)/(1 - sin x)?
Which identity is used when rewriting tan^(-1)(cos x)/(1 - sin x)?
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What relationship holds true based on the simplification of tan^(-1)(cos^2(x) - sin^2(x))?
What relationship holds true based on the simplification of tan^(-1)(cos^2(x) - sin^2(x))?
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What final form does tan^(-1)(1/(1 - sin x)) achieve following the simplifications shown?
What final form does tan^(-1)(1/(1 - sin x)) achieve following the simplifications shown?
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In the expression tan^(-1)(cos(x)/ (1 - sin(x))), which of the following represents the denominator correctly?
In the expression tan^(-1)(cos(x)/ (1 - sin(x))), which of the following represents the denominator correctly?
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What is the value of the expression tan^(-1)(1/(sin x + cos x)) equal to after manipulation?
What is the value of the expression tan^(-1)(1/(sin x + cos x)) equal to after manipulation?
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Which of the following correctly represents the conversion from sec^(-1)(x) to a cotangent form?
Which of the following correctly represents the conversion from sec^(-1)(x) to a cotangent form?
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What is the simplified outcome of cos^2(x) + sin^2(x) as described in the content?
What is the simplified outcome of cos^2(x) + sin^2(x) as described in the content?
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What is the principal value of an inverse trigonometric function?
What is the principal value of an inverse trigonometric function?
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Which of the following correctly expresses the relationship between y = sin^(-1)(x) and x?
Which of the following correctly expresses the relationship between y = sin^(-1)(x) and x?
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Who is credited with providing formulae to find the sine values for angles greater than 90°?
Who is credited with providing formulae to find the sine values for angles greater than 90°?
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In the equation sin(sin^(-1)(x)) = x, what can be concluded about the inputs?
In the equation sin(sin^(-1)(x)) = x, what can be concluded about the inputs?
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Which mathematician suggested the symbols sin^(-1)(x) and cos^(-1)(x) for arc sine and arc cosine functions?
Which mathematician suggested the symbols sin^(-1)(x) and cos^(-1)(x) for arc sine and arc cosine functions?
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What was the primary method used by Thales to determine the height of the pyramid?
What was the primary method used by Thales to determine the height of the pyramid?
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Which of the following historical figures is associated with the first significant study of trigonometry?
Which of the following historical figures is associated with the first significant study of trigonometry?
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What is the significance of the siddhantas in relation to modern trigonometric functions?
What is the significance of the siddhantas in relation to modern trigonometric functions?
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Study Notes
Inverse Trigonometric Functions
- Inverse trigonometric functions exist for functions that are one-to-one and onto.
- Trigonometric functions are not one-to-one over their natural domains and ranges, so their inverses do not exist initially.
- Restrictions on the domains and ranges of trigonometric functions are necessary to create inverses.
- Inverse trigonometric functions are essential in calculus to define integrals.
- They are also used in science and engineering.
Basic Concepts
- Trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant are defined.
- Their domains are specified (e.g., sine: R → [-1, 1]).
- Inverse functions are denoted using the superscript -1 (e.g., sin⁻¹).
Inverse Trigonometric Functions
- The domain of a trigonometric function becomes the range of its inverse.
- The range of a trigonometric function becomes the domain of its inverse.
- There exists a principal value branch for each inverse function.
Properties of Inverse Trigonometric Functions
- Inverse trigonometric functions have properties similar to other inverse functions.
- They are valuable in specific contexts within their domains.
- Examples demonstrating usage of these functions are included, in multiple ways across several examples.
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Description
Test your understanding of inverse trigonometric functions and their applications in calculus, science, and engineering. This quiz covers the essential concepts, including domain restrictions and properties of these functions. Challenge yourself with questions about sine, cosine, and their inverses.
