Trigonometric Identities Quiz

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.