PreCalculus Course Outcome 4: Inverse Trigonometric Functions
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Questions and Answers

What condition must a function meet to have an inverse?

  • It must be defined on a closed interval.
  • It must have a finite range.
  • It must be continuous everywhere.
  • It must be a one-to-one function. (correct)

    • Which of the following statements is true regarding inverses of trigonometric functions?

  • Trigonometric functions can only have inverses if their domains are restricted. (correct)
  • All trigonometric functions have natural inverses.
  • Inverse trigonometric functions have the same domain as their original functions.
  • Only sine functions can be inverted.

    • What is the range of the inverse sine function, $y = ext{sin}^{-1}(x)$?

  • $[- rac{eta}{2}, rac{eta}{2}]$ for $-1 eq x eq 0$
  • $[- rac{eta}{2}, rac{eta}{2}]$
  • $[- rac{eta}{2}, rac{eta}{2}]$ for $-1 eq x eq 1$ (correct)
  • $[0, eta]$

    • For the inverse cosine function $y = ext{cos}^{-1}(x)$, which is true about its range?

    <p>It is $[0, eta]$.</p> Signup and view all the answers

    What is the domain of the inverse sine function $y = ext{sin}^{-1}(x)$?

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

    Which graph accurately represents the inverse sine function?

    <p>It is symmetric about the line $y = x$.</p> Signup and view all the answers

    Which condition applies to the cosine function for it to be defined as an inverse?

    <p>The range must be limited to $[-1, 1]$.</p> Signup and view all the answers

    When considering the properties of the inverse sine and cosine functions, what similarity can be observed?

    <p>Both have a domain of $[-1, 1]$.</p> Signup and view all the answers

    Study Notes

    PreCalculus Course Outcome 4: Lesson 1 - Inverse Trigonometric Functions

    • Inverse functions are defined for one-to-one functions. Trigonometric functions aren't one-to-one, but their domains can be restricted to create one-to-one functions.

    Inverse Trigonometric Functions

    • If f(x) is one-to-one with domain A and range B, then its inverse f-1 has domain B and range A, defined by f-1(x) → f(y) = x

    Inverse Trigonometric Functions

    • In order for a function to have an inverse, it must be one-to-one. Trigonometric functions are not one-to-one, so they do not have inverses. However, it is possible to restrict trigonometric functions' domains to make them one-to-one.

    Inverse Sine Function and its Graph

    • y = sin-1 x if and only if x = sin y, where -1 ≤ x ≤ 1 and -π/2 ≤ y ≤ π/2.
    • Domain: [-1, 1]
    • Range: [-π/2, π/2]

    Inverse Cosine Function and its Graph

    • y = cos-1 x if and only if x = cos y, where -1 ≤ x ≤ 1 and 0 ≤ y ≤ π.
    • Domain: [-1, 1]
    • Range: [0, π]

    Inverse Tangent Function and its Graph

    • y = tan-1 x if and only if x = tan y, where -∞ < x < ∞ and -π/2 < y < π/2.
    • Domain: (-∞, ∞)
    • Range: (-π/2, π/2)

    Inverse Cotangent Function and its Graph

    • y = cot-1 x if and only if x = cot y, where -∞ < x < ∞ and 0 < y < π.
    • Domain: (-∞, ∞)
    • Range: (0, π)

    Inverse Secant Function and its Graph

    • y = sec-1 x if and only if x = sec y, where x ≤ -1 or x ≥ 1 and 0 ≤ y ≤ π, y ≠ π/2.
    • Domain: (-∞, -1] ∪ [1, ∞)
    • Range: [0, π/2) ∪ (π/2, π]

    Inverse Cosecant Function and its Graph

    • y = csc-1 x if and only if x = csc y, where x ≤ -1 or x ≥ 1 and -π/2 ≤ y ≤ π/2, y ≠ 0.
    • Domain: (-∞, -1] ∪ [1, ∞)
    • Range: [-π/2, 0) ∪ (0, π/2]

    Domain and Range of Inverse Functions

    • A table showing the domain and range of the inverse trigonometric functions.

    Evaluating Inverse Trigonometric Functions

    • When evaluating inverse trigonometric functions, look for the angle whose trigonometric function equals the given value. The angle must be in the range of the inverse function.

    Evaluating Inverse Trigonometric Functions (Special Angles)

    • Using a unit circle to help determine the correct angle. Includes diagrams of unit circles.

    Evaluating Inverse Trigonometric Functions (Examples)

    • Examples showing how to evaluate inverse trigonometric functions, including examples with special angles.

    Compositions of Functions

    • Inverse properties of functions. The inverse property does not apply for all values of x and y, but applies within the specified domain.

    Compositions of Functions (Examples)

    • Applying inverse properties to evaluate expressions.

    Examples (ALEKS)

    • Solved examples of inverse trigonometric function problems using specific input values.

    Solutions (Detailed Steps)

    • Specific examples with step-by-step solutions demonstrate evaluating and manipulating the different trigonometric identities.

    Example (Evaluation in Terms of 'u')

    • Example showcasing the evaluation of trigonometric terms involving sine, using a given variable u.

    Expanded Table of Special and Quadrantal Angles

    • A comprehensive table listing sine, cosine, tangent, cosecant, secant, and cotangent values for specific angles (in degrees and radians).

    Answers to Examples

    • Solutions and calculated answers for previously posed examples.

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    PreCalculus Course Outcome 4 - Inverse Trigonometric Functions PDF

    Description

    Explore the concepts of inverse trigonometric functions in this lesson. Learn how trigonometric functions can be manipulated to create one-to-one functions, allowing for the definition of their inverses. Understand the domains and ranges of the inverse sine and cosine functions, including their graphs.

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