Trigonometry Identities Lecture Notes

Introduction

• Overview of common trigonometry identities.
• Useful for beginners or exam preparation.

Right Triangle Basics

• Triangle components: opposite side, adjacent side, hypotenuse.

Trigonometric Ratios

• SOHCAHTOA:
  • Sine (sin): Opposite / Hypotenuse
  • Cosine (cos): Adjacent / Hypotenuse
  • Tangent (tan): Opposite / Adjacent

Reciprocal Identities

• Cosecant (csc): 1 / sin
• Secant (sec): 1 / cos
• Cotangent (cot): 1 / tan

Example Problem

• For a 3-4-5 triangle:
  • sin(θ) = 4/5
  • cos(θ) = 3/5
  • tan(θ) = 4/3
  • Reciprocal Examples:
    • csc(θ) = 5/4
    • sec(θ) = 5/3
    • cot(θ) = 3/4

Quotient Identities

• tan(θ) = sin(θ) / cos(θ)
• cot(θ) = cos(θ) / sin(θ)

Pythagorean Identities

• sin²(θ) + cos²(θ) = 1
• 1 + cot²(θ) = csc²(θ)
• 1 + tan²(θ) = sec²(θ)

Even and Odd Functions

• Even functions: cos(θ), sec(θ)
• Odd functions: sin(θ), csc(θ), tan(θ), cot(θ)

Cofunction Identities

• cos(θ) = sin(90° - θ)
• sec(θ) = csc(90° - θ)
• tan(θ) = cot(90° - θ)

Angle Sum and Difference

• Sine: sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
• Cosine: cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
• Tangent: (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β))

Double Angle Identities

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
• tan(2θ) = 2tan(θ) / (1 - tan²(θ))

Half Angle Identities

• sin(θ/2) = ±√((1 - cos(θ))/2)
• cos(θ/2) = ±√((1 + cos(θ))/2)
• tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))

Power Reducing Formulas

• sin²(θ) = (1 - cos(2θ)) / 2
• cos²(θ) = (1 + cos(2θ)) / 2

Product-to-Sum and Sum-to-Product Formulas

• Product-to-Sum:
  • sin(α)sin(β) = 1/2[cos(α-β) - cos(α+β)]
  • cos(α)cos(β) = 1/2[cos(α-β) + cos(α+β)]
  • sin(α)cos(β) = 1/2[sin(α+β) + sin(α-β)]
• Sum-to-Product:
  • sin(α) + sin(β) = 2sin[(α+β)/2]cos[(α-β)/2]
  • cos(α) + cos(β) = 2cos[(α+β)/2]cos[(α-β)/2]

Law of Sines

• sin(A)/a = sin(B)/b = sin(C)/c

Law of Cosines

• c² = a² + b² - 2ab cos(C)

Area of a Triangle

• Area = 1/2 ab sin(C)

Law of Tangents (Less Common)

• (a-b)/(a+b) = (tan[(A-B)/2]) / (tan[(A+B)/2])

Conclusion

• Important trigonometric identities for courses and exams.