ট্রিগনোমেট্রি: সাইন, কসাইন, ট্যানজেন্ট এবং আইডেন্টিটি

Trigonometry

Sine, Cosine, and Tangent

• Sine (sin): ratio of the opposite side to the hypotenuse in a right-angled triangle
  • sin(A) = opposite side / hypotenuse
• Cosine (cos): ratio of the adjacent side to the hypotenuse in a right-angled triangle
  • cos(A) = adjacent side / hypotenuse
• Tangent (tan): ratio of the opposite side to the adjacent side in a right-angled triangle
  • tan(A) = opposite side / adjacent side

Trigonometric Equations

• Types of trigonometric equations:
  • Simple trigonometric equations: involve a single trig function (e.g., sin(x) = 1/2)
  • Compound trigonometric equations: involve multiple trig functions (e.g., sin(x) + cos(x) = 1)
• Solving trigonometric equations:
  • Use inverse trig functions (e.g., arcsin, arccos, arctan) to find solutions
  • Use algebraic methods (e.g., substitution, factoring) to simplify and solve equations

Inverse Trigonometric Functions

• Inverse sine (arcsin or sin^(-1)): returns the angle whose sine is a given value
  • arcsin(x) = θ, where sin(θ) = x
• Inverse cosine (arccos or cos^(-1)): returns the angle whose cosine is a given value
  • arccos(x) = θ, where cos(θ) = x
• Inverse tangent (arctan or tan^(-1)): returns the angle whose tangent is a given value
  • arctan(x) = θ, where tan(θ) = x

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))