Inverse Trigonometric Functions Quiz

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Questions and Answers

What is the domain of the sine function?

  • The open interval (-1, 1)
  • The closed interval [-1, 1]
  • All positive real numbers
  • The set of real numbers (correct)

What is the range of the sine function?

  • The open interval (-1, 1)
  • The closed interval [-1, 1] (correct)
  • The set of real numbers
  • All positive real numbers

The inverse of the (restricted) sine function $y = \text{sinx}$, where the domain is restricted to the closed interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$, is called the ____ or arcsine function.

inverse sine function

If $y = \text{sin}^{-1}x$ or $y = \text{arcsin }x$, which of the following is true?

<p>$-1 \leq x \leq 1$ and $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$ (C)</p> Signup and view all the answers

What is the domain of $y = \text{arcsin }x$?

<p>$[-1, 1]$</p> Signup and view all the answers

What is the range of $y = \text{arcsin }x$?

<p>$[-\frac{\pi}{2}, \frac{\pi}{2}]$</p> Signup and view all the answers

Find the exact value of $\text{sin}^{-1}(\frac{1}{2})$.

<p>$\frac{\pi}{6}$</p> Signup and view all the answers

Find the exact value of $\text{sin}^{-1}(-\frac{\sqrt{3}}{2})$.

<p>$-\frac{\pi}{3}$</p> Signup and view all the answers

The development of the inverse cosine function follows similarly from that of the ____ function.

<p>inverse sine</p> Signup and view all the answers

If $y = \text{cos}^{-1}x$ or $y = \text{arccos }x$, which of the following is true?

<p>$-1 \leq x \leq 1$ and $0 \leq y \leq \pi$ (A)</p> Signup and view all the answers

Find the angle $\theta$, $0 \leq y \leq \pi$, whose cosine value equals $\frac{\sqrt{2}}{2}$.

<p>$\frac{\pi}{4}$</p> Signup and view all the answers

What is the angle between 0 and $\pi$, inclusive, whose cosine value is $-\frac{1}{2}$?

<p>$\frac{2\pi}{3}$</p> Signup and view all the answers

What is the domain of the inverse tangent function?

<p>All real number</p> Signup and view all the answers

What is the range of the inverse tangent function?

<p>$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$</p> Signup and view all the answers

Flashcards

Inverse Sine Function

The inverse function of sine, denoted as y = arcsin x, defined for x in [-1, 1] with range [-π/2, π/2].

Domain of arcsin

The set of x-values for arcsin, which is [-1, 1].

Range of arcsin

The set of output angles for arcsin, which is [-π/2, π/2].

Inverse Cosine Function

The inverse function of cosine, denoted as y = arccos x, defined for x in [-1, 1] with range [0, π].

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Domain of arccos

The set of x-values for arccos, which is [-1, 1].

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Range of arccos

The set of output angles for arccos, which is [0, π].

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Inverse Tangent Function

The inverse function of tangent, denoted as y = arctan x, defined for all real numbers with range (-π/2, π/2).

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Domain of arctan

All real numbers ℝ.

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Range of arctan

The output angles for arctan, which is (-π/2, π/2).

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General Value of arcsin

arcsin(x) = y + 2nπ or y = π - arcsin(x) for any integer n.

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General Value of arccos

arccos(x) = y + 2nπ where n is any integer.

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Inverse Properties of sine

sin(sin^(-1)(x)) = x for x in [-1, 1].

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Inverse Properties of cosine

cos(cos^(-1)(x)) = x for x in [-1, 1].

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Inverse Properties of tangent

tan(tan^(-1)(x)) = x for all real x.

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Sine Double Angle Identity

sin(2θ) = 2sin(θ)cos(θ).

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Tangent Difference Identity

tan(α - β) = (tan α - tan β) / (1 + tan α tan β).

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Properties of arctan

tan(arctan(x)) = x, valid for any real x.

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Arctsine and arccosine relationship

arccos(x) = π/2 - arcsin(x) for 0 ≤ x ≤ 1.

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Arccosine and arcsine relationship

arcsin(x) = π/2 - arccos(x) for 0 ≤ x ≤ 1.

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Inverse properties of cosecant

csc(y) = x implies y = csc^(-1)(x).

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Inverse properties of secant

sec(y) = x implies y = sec^(-1)(x).

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Inverse properties of cotangent

cotangent function geometric relationships.

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Graph of arcsin

A curve from (-1, -π/2) to (1, π/2).

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Graph of arccos

A curve from (-1, π) to (1, 0).

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Graph of arctan

A curve approaching -π/2 and π/2 asymptotically.

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Graph of arccos relationships

Graphs are complementary to sine and tangent inverse characteristics.

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Study Notes

Inverse Trigonometric Functions

  • Inverse trigonometric functions are used to find angles from trigonometric ratios.
  • The sine function is not one-to-one, so a restricted domain is needed for an inverse.
  • The restricted domain for arcsin (inverse sine) is [-π/2, π/2].
  • The restricted domain for arccos (inverse cosine) is [0, π].
  • The restricted domain for arctan (inverse tangent) is (-π/2, π/2).

Inverse Sine Function

  • The inverse sine function (arcsin), denoted as sin⁻¹(x) or arcsin(x), is the inverse of the sine function.
  • The input is a ratio, and the output is the angle (in radians).
  • The domain of arcsin(x) is [-1, 1].
  • The range of arcsin(x) is [-π/2, π/2].

Inverse Cosine Function

  • The inverse cosine function (arccos), denoted as cos⁻¹(x) or arccos(x), is the inverse of the restricted cosine function.
  • The input is a ratio, and the output is the angle (in radians).
  • The domain of arccos(x) is [-1, 1].
  • The range of arccos(x) is [0, π].

Inverse Tangent Function

  • The inverse tangent function (arctan), denoted as tan⁻¹(x) or arctan(x), is the inverse of the restricted tangent function.
  • The input is a ratio, and the output is the angle (in radians).
  • The domain of arctan(x) is all real numbers.
  • The range of arctan(x) is (-π/2, π/2).

Inverse Cotangent Function

  • The inverse cotangent function (arccot), denoted as cot⁻¹(x) or arccot(x), is the inverse of the restricted cotangent function.
  • The input is a ratio, and the output is the angle (in radians).
  • The domain of arccot(x) is all real numbers.
  • The range of arccot(x) is (0, π).

Inverse Secant Function

  • The inverse secant function (arcsec), denoted as sec⁻¹(x) or arcsec(x), is the inverse of the restricted secant function.
  • The input is a ratio, and the output is the angle (in radians).
  • The domain of arcsec(x) is (-∞, -1] ∪ [1, ∞).
  • The range of arcsec(x) is [0, π/2) ∪ (π/2, π].

Inverse Cosecant Function

  • The inverse cosecant function (arccsc), denoted as csc⁻¹(x) or arccsc(x), is the inverse of the restricted cosecant function.
  • The input is a ratio, and the output is the angle (in radians).
  • The domain of arccsc(x) is (-∞, -1] ∪ [1, ∞).
  • The range of arccsc(x) is [-π/2, 0) ∪ (0, π/2].

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