Trigonometry Problem: Solving tan^-1(-3)
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Questions and Answers

What is the range of the function f(x) = cos(tan⁻¹(x))?

  • [0, 1]
  • [-1, 0]
  • [-1, 1]
  • (0, 1] (correct)

    • What is the property of cos⁻¹(x) used in the given inequalities?

  • Cos⁻¹(x) is an increasing function
  • Cos⁻¹(x) is an odd function
  • Cos⁻¹(x) is a decreasing function (correct)
  • Cos⁻¹¹(x) is a periodic function

    • What is the graph of the function y = cos(x) like in the interval [-π/2, π/2]?

  • A constant function
  • A straight line
  • A decreasing curve
  • An increasing curve (correct)

    • What is the range of tan⁻¹(x) in the given solution?

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

    What is the purpose of multiplying both sides of the inequality by 3?

    <p>To eliminate the fraction</p> Signup and view all the answers

    What is the domain of the function f(x) = cos(tan⁻¹(x))?

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

    What is the range of the function f(x) = sin x?

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

    What is the domain of the function f(x) = cos^{-1} x?

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

    What is the value of sin^{-1} (1/2)?

    <p>π/6</p> Signup and view all the answers

    What is the property of the function f(x) = tan^{-1} x?

    <p>tan^{-1} x = -tan^{-1} (-x)</p> Signup and view all the answers

    What is the graph of the function f(x) = sin^{-1} x?

    <p>A continuous curve from -π/2 to π/2</p> Signup and view all the answers

    What is the range of the function f(x) = tan^{-1} x?

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

    What is the value of cos^{-1} (√3/2)?

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

    What is the property of the function f(x) = cos^{-1} x?

    <p>cos^{-1} x = -cos^{-1} (-x)</p> Signup and view all the answers

    What is the domain of the function f(x) = tan x?

    <p>(-∞, ∞)</p> Signup and view all the answers

    What is the graph of the function f(x) = cos x?

    <p>A continuous curve from -π to π</p> Signup and view all the answers

    What is the range of the function $f(x) = an x$?

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

    What is the value of $ an^{-1}(-3)$?

    <p>$-rac{π}{3}$</p> Signup and view all the answers

    What is the value of $ an^{-1}(1) + rac{π}{2}$?

    <p>$rac{3π}{2}$</p> Signup and view all the answers

    What is the value of $ an^{-1}(1) + rac{π}{2} + rac{π}{6}$?

    <p>$rac{5π}{3}$</p> Signup and view all the answers

    What is the value of $ an^{-1}(1) + rac{3π}{2} + rac{π}{6}$?

    <p>$rac{7π}{3}$</p> Signup and view all the answers

    Study Notes

    Inverse Trigonometric Functions

    Domain, Range & Graph of ITF

    • tan:
      • Restricted Domain: (-π, π/2) ∪ (π/2, π)
      • Range: ℝ
    • cot:
      • Restricted Domain: (0, π)
      • Range: ℝ
    • sec:
      • Restricted Domain: (0, π) - π/2 - π
      • Range: (-∞, -1] ∪ [1, ∞)
    • cosec:
      • Restricted Domain: (-π/2, 0) ∪ (0, π/2)
      • Range: (-∞, -1] ∪ [1, ∞)

    Examples

    • cot⁻¹(0) = π/2
    • cot⁻¹(1) = π/4
    • cot⁻¹(-1) = 3π/4

    Properties of Inverse Trigonometric Functions

    • sin⁻¹(-x) = -sin⁻¹(x)
    • cos⁻¹(-x) = π - cos⁻¹(x)
    • tan⁻¹(-x) = -tan⁻¹(x)
    • cot⁻¹(-x) = π - cot⁻¹(x)
    • sec⁻¹(-x) = π - sec⁻¹(x)
    • cosec⁻¹(-x) = -cosec⁻¹(x)

    Graphs of Inverse Trigonometric Functions

    • y = sin⁻¹(x)
      • Domain: [-1, 1]
      • Range: [-π/2, π/2]
    • y = cos⁻¹(x)
      • Domain: [-1, 1]
      • Range: [0, π]
    • y = tan⁻¹(x)
      • Domain: ℝ
      • Range: (-π/2, π/2)

    Additional Examples

    • Find the value of tan⁻¹(3) and tan⁻¹(-3)
      • Solution: use the properties of inverse trigonometric functions to find the values.

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    Description

    Solve for θ when tangent inverse of -3 is equal to θ, where tangent of θ is equal to -3. Trigonometry problem involving inverse tangent function.

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