Trigonometric Identities Quiz
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Questions and Answers

What is the value of sin^2(A) + cos^2(A) according to the Pythagorean Identity?

  • 1 (correct)
  • -1
  • 0
  • 2
  • What is the formula for sin(A + B)?

  • sin(A) - cos(B)
  • sin(A) + cos(B)
  • sin(A)cos(B) + cos(A)sin(B) (correct)
  • sin(A)cos(B) - cos(A)sin(B)
  • What is the condition for a function f(x) to be continuous at a point x=a?

  • f(a) is undefined
  • f(a) is defined, lim x→a f(x) exists, and lim x→a f(x) = f(a) (correct)
  • lim x→a f(x) exists and lim x→a f(x) ≠ f(a)
  • f(a) is defined and lim x→a f(x) exists
  • What type of discontinuity occurs when the left and right limits exist but are not equal?

    <p>Jump Discontinuity</p> Signup and view all the answers

    What is the result of the Sum Theorem for two continuous functions?

    <p>The sum is continuous</p> Signup and view all the answers

    What is the formula for cos(2A)?

    <p>cos^2(A) - sin^2(A)</p> Signup and view all the answers

    What is the result of the Intermediate Value Theorem?

    <p>There exists a value c in [a, b] such that f(c) = k</p> Signup and view all the answers

    What is the formula for tan(2A)?

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

    What is the result of the Power Reduction Formula for sin^2(A)?

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

    Study Notes

    Trigonometric Identities

    • Pythagorean Identity: sin^2(A) + cos^2(A) = 1
    • Sum and Difference Formulas:
      • sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
      • cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
      • tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
    • Double Angle Formulas:
      • sin(2A) = 2sin(A)cos(A)
      • cos(2A) = cos^2(A) - sin^2(A)
      • tan(2A) = (2tan(A)) / (1 - tan^2(A))
    • Power Reduction Formulas:
      • sin^2(A) = (1 - cos(2A)) / 2
      • cos^2(A) = (1 + cos(2A)) / 2

    Continuity

    • Definition of Continuity: A function f(x) is continuous at a point x=a if:
      1. f(a) is defined
      2. lim x→a f(x) exists
      3. lim x→a f(x) = f(a)
    • Types of Discontinuity:
      • Removable Discontinuity: A discontinuity that can be removed by redefining the function at a single point.
      • Jump Discontinuity: A discontinuity where the left and right limits exist but are not equal.
      • Infinite Discontinuity: A discontinuity where the function approaches infinity or negative infinity.
    • Theorems:
      • The Sum Theorem: The sum of two continuous functions is continuous.
      • The Product Theorem: The product of two continuous functions is continuous.
      • The Chain Rule Theorem: The composite of two continuous functions is continuous.
    • Intermediate Value Theorem: If a function f(x) is continuous on the closed interval [a, b] and k is any value between f(a) and f(b), then there exists a value c in [a, b] such that f(c) = k.

    Trigonometric Identities

    • Pythagorean Identity: Relates the sine and cosine of an angle, stating that the sum of their squares is always 1.

    Trigonometric Formulas

    • Sum and Difference Formulas: Allow calculation of sine, cosine, and tangent values for the sum or difference of two angles.
      • Sine Formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
      • Cosine Formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
      • Tangent Formula: tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
    • Double Angle Formulas: Relate the sine, cosine, and tangent values of an angle to those of its double angle.
      • Sine Formula: sin(2A) = 2sin(A)cos(A)
      • Cosine Formula: cos(2A) = cos^2(A) - sin^2(A)
      • Tangent Formula: tan(2A) = (2tan(A)) / (1 - tan^2(A))
    • Power Reduction Formulas: Simplify trigonometric expressions involving squared sine or cosine values.
      • Sine Formula: sin^2(A) = (1 - cos(2A)) / 2
      • Cosine Formula: cos^2(A) = (1 + cos(2A)) / 2

    Continuity of Functions

    Definition and Types of Discontinuity

    • Continuity Definition: A function is continuous at a point if it is defined, has a limit, and the limit equals the function value.
    • Types of Discontinuity: Removable, Jump, and Infinite discontinuities are the three main categories of discontinuities.

    Theorems on Continuous Functions

    • The Sum Theorem: The sum of two continuous functions is continuous.
    • The Product Theorem: The product of two continuous functions is continuous.
    • The Chain Rule Theorem: The composite of two continuous functions is continuous.

    Intermediate Value Theorem

    • Statement: If a continuous function takes values f(a) and f(b) on a closed interval [a, b], then it must take any value k between f(a) and f(b) at some point c within the interval.

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    Test your knowledge of trigonometric identities including Pythagorean, sum and difference, double angle, and power reduction formulas.

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