Inverse Trigonometric Functions
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Questions and Answers

What is the range of the function sin–1?

  • [0, π]
  • [–π/2, π/2] (correct)
  • [0, 2π]
  • [–1, 1]

Which of the following functions is defined for the range R – (–1, 1)?

  • tan–1
  • cosec–1 (correct)
  • sin–1
  • cos–1

What is the correct interpretation of sin–1x?

  • The same as (sin x)–1
  • The angle whose sine is x (correct)
  • The reciprocal of sin x
  • The value of sin x at x

What is the principal value of cot–1(3)?

<p>π/3 (B)</p> Signup and view all the answers

Which function has a range of (0, π)?

<p>cot–1 (C)</p> Signup and view all the answers

What is the principal value range for the sec–1 function?

<p>[0, π/2) ∪ (π/2, π] (A)</p> Signup and view all the answers

For which value is sin–1(1/2) equal to?

<p>π/3 (B)</p> Signup and view all the answers

Which of the following statements is false regarding inverse trigonometric functions?

<p>cot–1 is only defined for positive values. (A)</p> Signup and view all the answers

Which of the following ranges belongs to the function sin–1?

<p>(−π/2, π/2) (D)</p> Signup and view all the answers

What is the result of sin–1(−1)?

<p>−$ rac{ heta}{2}$ (D)</p> Signup and view all the answers

If sin–1 x = y, which statement is true about y?

<p>$− rac{ heta}{2} &lt; y &lt; rac{ heta}{2}$ (C)</p> Signup and view all the answers

Which expression evaluates to π/3?

<p>cos–1(1/2) (C)</p> Signup and view all the answers

What is the sum of tan–1(1) and cos–1(1/2)?

<p>$ rac{π}{2}$ (A)</p> Signup and view all the answers

What is the value of tan–1(−3) given the properties of inverse functions?

<p>−$ rac{π}{3}$ (B)</p> Signup and view all the answers

What is the valid range for x when using the equation $y = sin^{-1} x$?

<p>x ∈ [-1, 1] (B)</p> Signup and view all the answers

Which identity is correct for the equation $sin^{-1}(sin y)$?

<p>sin^{-1}(sin y) = y, y ∈ [-π/2, π/2] (D)</p> Signup and view all the answers

What is the result of $sin^{-1}(2sinθcosθ)$?

<p>2θ (C)</p> Signup and view all the answers

When working with inverse trigonometric functions, which statement is correct about their domains?

<p>They have specific domains for certain values of x. (B)</p> Signup and view all the answers

For the relation $sin^{-1}(2x(1-x^2)) = 2sin^{-1} x$, what is the condition for x?

<p>0 ≤ x ≤ 1 (C)</p> Signup and view all the answers

In the equation $sin^{-1}(2x(1-x^2)) = 2cos^{-1} x$, what is the required range for x?

<p>0 ≤ x ≤ 1 (D)</p> Signup and view all the answers

What is the expression for $tan^{-1}$ in its simplest form as mentioned?

<p>Not provided in the content. (B)</p> Signup and view all the answers

What is the equivalent relationship when $y = cos^{-1}x$?

<p>cos y = x (C)</p> Signup and view all the answers

What is the range of the function y = sin–1 x?

<p>[−π/2, π/2] (D)</p> Signup and view all the answers

Which of the following equations relates to y = tan–1 x?

<p>y = cot–1 x (C)</p> Signup and view all the answers

What is the domain of the function y = cosec–1 x?

<p>R – (−1, 1) (D)</p> Signup and view all the answers

For the equation sin–1(1 – x) – 2 sin–1 x = π/2, what is one possible value of x?

<p>1/2 (A)</p> Signup and view all the answers

Which statement correctly describes the inverse trigonometric function y = cos–1 x?

<p>Domain: [−1, 1]; Range: [0, π] (A)</p> Signup and view all the answers

What is the relationship between sin(tan–1 x) and x for |x| < 1?

<p>sin(tan–1 x) = x/(1 + x^2) (C)</p> Signup and view all the answers

Which of the following accurately describes the output range of y = sec–1 x?

<p>[0, π] – {π/2} (C)</p> Signup and view all the answers

What is the simplest form of cot(2/(x - 1)), where x > 1?

<p>sec^(-1)(x) (A)</p> Signup and view all the answers

Which identity is used when rewriting tan^(-1)(cos x)/(1 - sin x)?

<p>tan^(-1)(1 - sin^2(x)) (C)</p> Signup and view all the answers

What relationship holds true based on the simplification of tan^(-1)(cos^2(x) - sin^2(x))?

<p>tan^(-1)(cos x - sin x) (D)</p> Signup and view all the answers

What final form does tan^(-1)(1/(1 - sin x)) achieve following the simplifications shown?

<p>tan^(-1)(cot(x/2)) (B)</p> Signup and view all the answers

In the expression tan^(-1)(cos(x)/ (1 - sin(x))), which of the following represents the denominator correctly?

<p>1 - sin(x) (A)</p> Signup and view all the answers

What is the value of the expression tan^(-1)(1/(sin x + cos x)) equal to after manipulation?

<p>tan^(-1)(tan(x/2)) (B)</p> Signup and view all the answers

Which of the following correctly represents the conversion from sec^(-1)(x) to a cotangent form?

<p>cot(θ) (D)</p> Signup and view all the answers

What is the simplified outcome of cos^2(x) + sin^2(x) as described in the content?

<p>1 (C)</p> Signup and view all the answers

What is the principal value of an inverse trigonometric function?

<p>The value that lies in its principal value branch (A)</p> Signup and view all the answers

Which of the following correctly expresses the relationship between y = sin^(-1)(x) and x?

<p>x = sin(y) (B)</p> Signup and view all the answers

Who is credited with providing formulae to find the sine values for angles greater than 90°?

<p>Bhaskara I (A)</p> Signup and view all the answers

In the equation sin(sin^(-1)(x)) = x, what can be concluded about the inputs?

<p>x must lie within the range of [-1, 1] (A)</p> Signup and view all the answers

Which mathematician suggested the symbols sin^(-1)(x) and cos^(-1)(x) for arc sine and arc cosine functions?

<p>Sir John F.W. Herschel (D)</p> Signup and view all the answers

What was the primary method used by Thales to determine the height of the pyramid?

<p>Using similar triangles (D)</p> Signup and view all the answers

Which of the following historical figures is associated with the first significant study of trigonometry?

<p>Aryabhata (B)</p> Signup and view all the answers

What is the significance of the siddhantas in relation to modern trigonometric functions?

<p>They provided foundational concepts for sine functions (D)</p> Signup and view all the answers

Flashcards

What is the inverse sine function (sin⁻¹)?

The inverse sine function, denoted as sin⁻¹(x) or arcsin(x), is a mathematical function that returns the angle whose sine is x. It's essentially the opposite operation of the sine function. For example, sin⁻¹(1/2) = π/6 because sin(π/6) = 1/2.

What is the domain of the inverse sine function (sin⁻¹)?

The domain of the inverse sine function is the closed interval [-1, 1]. This means that the input 'x' must be a value between -1 and 1, inclusive. It's because the sine function's range is also within this interval.

What is the range of the inverse sine function (sin⁻¹)?

The range of the inverse sine function is the closed interval [-π/2, π/2]. This means that the output of the function will always be an angle between -π/2 and π/2, inclusive.

What is the inverse cosine function (cos⁻¹)?

The inverse cosine function, denoted as cos⁻¹(x) or arccos(x), is a mathematical function that returns the angle whose cosine is x. It's the reverse operation of the cosine function.

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What is the domain of the inverse cosine function (cos⁻¹)?

The domain of the inverse cosine function is the closed interval [-1, 1]. This means that the input 'x' must be a value between -1 and 1, inclusive. It's because the cosine function's range is also within this interval.

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What is the range of the inverse cosine function (cos⁻¹)?

The range of the inverse cosine function is the closed interval [0, π]. This means that the output of the function will always be an angle between 0 and π, inclusive.

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What is the inverse cosecant function (cosec⁻¹)?

The inverse cosecant function, denoted as cosec⁻¹(x) or arccsc(x), is a mathematical function that returns the angle whose cosecant is x. It's the opposite operation of the cosecant function, where cosec(x) = 1/sin(x).

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cot–1 Range

The range of principal value branch of cot–1 is 0 to π (exclusive).

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Finding Principal Value of cot–1

Finding the principal value of cot–1 involves finding the angle in the specified range (0 to π) that corresponds to the given cotangent value.

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Inverse Sine (sin–1)

The inverse sine function (sin–1) takes a value between -1 and 1 and returns an angle (in radians) between -π/2 and π/2.

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Inverse Cosine (cos–1)

The inverse cosine function (cos–1) takes a value between -1 and 1 and returns an angle (in radians) between 0 and π.

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Inverse Cosecant (cosec–1)

The inverse cosecant function (cosec–1) takes a value and returns an angle (in radians) between -π/2 to π/2, excluding 0.

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Inverse Tangent (tan–1)

The inverse tangent function (tan–1) takes a value and returns an angle (in radians) between -π/2 and π/2.

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Inverse Secant (sec–1)

The inverse secant function (sec–1) takes a value and returns an angle (in radians) between 0 and π, excluding π/2.

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Inverse Cotangent (cot–1)

The inverse cotangent function (cot–1) takes a value and returns an angle (in radians) between 0 and π.

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What are inverse trigonometric functions?

The inverse trigonometric functions are the inverses of the trigonometric functions. They are used to find the angle that corresponds to a given trigonometric ratio.

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What is the principal value branch?

The principal value branch of an inverse trigonometric function is the range of values that the function can output, ensuring a unique output for each input.

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What is the inverse tangent function (tan⁻¹)?

The inverse tangent function, denoted as tan⁻¹(x) or arctan(x), returns the angle whose tangent is x.

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What is the domain of inverse trigonometric functions?

The domain of an inverse trigonometric function is the set of all possible input values that the function can accept. In the case of sin⁻¹, cos⁻¹, and tan⁻¹, the domain is typically restricted to ensure a unique output.

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What is the range of inverse trigonometric functions?

The range of an inverse trigonometric function is the set of all possible output values that the function can produce.

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When are inverse trigonometric identities true?

These results are valid only within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined. Some results may not be valid for all values of the domains of inverse trigonometric functions.

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What is the inverse tangent function?

The inverse tangent function, written as tan⁻¹(x) or arctan(x), is a mathematical function that returns the angle whose tangent is x. It's essentially the opposite operation of the tangent function. For example, tan⁻¹(1) = π/4 because tan(π/4) = 1.

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What's the domain of the inverse tangent function?

The domain of the inverse tangent function is all real numbers. This means you can input any real number 'x' into the function.

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What's the range of the inverse tangent function?

The range of the inverse tangent function is the open interval (-π/2, π/2). This means the output of the function will always be an angle between -π/2 and π/2, excluding both endpoints.

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What is the inverse cotangent function?

The inverse cotangent function, written as cot⁻¹(x) or arccot(x), is a mathematical function that returns the angle whose cotangent is x. It's the opposite operation of the cotangent function, where cot(x) = 1/tan(x).

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What's the domain of the inverse cotangent function?

The domain of the inverse cotangent function is all real numbers. This means you can input any real number 'x' into the function.

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What's the range of the inverse cotangent function?

The range of the inverse cotangent function is the open interval (0, π). This means the output of the function will always be an angle between 0 and π, excluding both endpoints.

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What is the inverse secant function?

The inverse secant function, written as sec⁻¹(x) or arcsec(x), is a mathematical function that returns the angle whose secant is x. It's the opposite operation of the secant function, where sec(x) = 1/cos(x).

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What is the inverse cosecant function?

The inverse cosecant function, written as cosec⁻¹(x) or arccsc(x), is a mathematical function that returns the angle whose cosecant is x. It's the opposite operation of the cosecant function, where cosec(x) = 1/sin(x).

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sin(sin⁻¹(x)) = x

For a given value of x, the expression sin(sin⁻¹(x)) will always equal x because the inverse sine function essentially reverses the effect of the sine function.

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Origins of the Sine Function

The name sine function originated from the Indian astronomical works called siddhantas, which introduced the sine function. This function has played a vital role in the development of trigonometry throughout history.

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Who are some notable Indian mathematicians who made contributions to trigonometry?

Aryabhata, Brahmagupta, Bhaskara I and II were some of the ancient Indian mathematicians who made significant contributions to trigonometry, laying the foundation for many key results and concepts.

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What are Thales' contributions to geometry?

Thales, a Greek mathematician, made notable contributions to geometry, particularly in height and distance problems. He is credited with using similar triangles to calculate the height of the Great Pyramid of Giza and the distance of ships at sea.

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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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Inverse Trigonometric Functions PDF

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.

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