Trigonometry (Copy)

10 Trigonometry – Cheat Sheet

10.1 Six Trigonometric Functions

• Primary:
  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent
• Reciprocal:
  • sec θ = 1 / cos θ
  • cosec θ = 1 / sin θ
  • cot θ = 1 / tan θ = cos θ / sin θ

10.2 Amplitude & Period

• Amplitude (a): max displacement from midline in y = a sin bx or y = a cos bx → amplitude = |a|
• Period (T):
  • Degrees: T = 360° / b
  • Radians: T = 2π / b
• Adding c → vertical shift (up/down)
• Horizontal shift if phase change present (not in this syllabus form)

10.3 Graphs

• y = a sin bx + c → sine wave, starts at midline
• y = a cos bx + c → cosine wave, starts at max (if a > 0)
• y = a tan bx + c → tangent graph, vertical asymptotes at x = (π/2b) + nπ/b (radians) or (90°/b) + n(180°/b) (degrees)
• Label all asymptotes for tan graphs

10.4 Trigonometric Identities

• sin² A + cos² A = 1
• sec² A = 1 + tan² A
• cosec² A = 1 + cot² A
• Reciprocal & quotient rules:
  • tan A = sin A / cos A
  • cot A = cos A / sin A

10.5 Solving Trigonometric Equations

• Reduce all expressions to sin, cos, tan if possible
• Use identities from 10.4 to simplify
• Solve basic equation for θ in one period, then add general solution:
  • Degrees: θ = θ₀ + n × 360° or θ = (180° – θ₀) + n × 360° (for sine)
  • Radians: θ = θ₀ + 2nπ or θ = (π – θ₀) + 2nπ
• Remember function signs in quadrants (CAST rule)

10.6 Proving Trigonometric Identities

• Work with one side only
• Convert all sec, cosec, cot to sin, cos, tan where possible
• Use:
  • 1 + tan² A = sec² A
  • 1 + cot² A = cosec² A
  • sin² A = 1 – cos² A
• Common tricks:
  • Multiply numerator and denominator by conjugate
  • Factorise expressions
  • Split fractions