Common Trigonometric Identities and Formulas

Introduction

• Overview of essential trigonometric identities
• Useful for beginners and for final exam preparation

Right Triangle Basics

• Angle theta, opposite side, adjacent side, hypotenuse (across 90° angle)

Soh Cah Toa

• Sine (Soh): sin(θ) = opposite / hypotenuse
• Cosine (Cah): cos(θ) = adjacent / hypotenuse
• Tangent (Toa): tan(θ) = opposite / adjacent

Reciprocal Identities

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

Example Problem: 3-4-5 Right Triangle

• Sine(θ): sin(θ) = 4 / 5
• Cosine(θ): cos(θ) = 3 / 5
• Tangent(θ): tan(θ) = 4 / 3
• Reciprocal Ratios:
  • 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
• Derivative Identities:
  • 1 + cot²(θ) = csc²(θ) (divide by sin²(θ))
  • 1 + tan²(θ) = sec²(θ) (divide by cos²(θ))

Even and Odd Functions

• Odd Functions:
  • sin(-θ) = -sin(θ)
  • tan(-θ) = -tan(θ)
  • csc(-θ) = -csc(θ)
  • cot(-θ) = -cot(θ)
• Even Functions:
  • cos(-θ) = cos(θ)
  • sec(-θ) = sec(θ)

Co-Function Identities

• cos(θ) = sin(90° - θ)
• sin(θ) = cos(90° - θ)
• csc(θ) = sec(90° - θ)
• sec(θ) = csc(90° - θ)
• cot(θ) = tan(90° - θ)
• tan(θ) = cot(90° - θ)
• Example: cos(30°) = sin(60°)

Double Angle Identities

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

Sum and Difference Identities

• 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 ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α) tan(β))

Power-Reducing Formulas

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

Product to Sum Formulas

• 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)]
• cos(α)sin(β) = 1/2 [sin(α + β) - sin(α - β)]

Sum to Product Formulas

• sin(α) + sin(β) = 2 sin((α + β)/2) cos((α - β)/2)
• sin(α) - sin(β) = 2 cos((α + β)/2) sin((α - β)/2)
• cos(α) + cos(β) = 2 cos((α + β)/2) cos((α - β)/2)
• cos(α) - cos(β) = -2 sin((α + β)/2) sin((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

• Using trig: (1/2)ab sin(C)
• Using Heron's Formula:
  • S = (a + b + c) / 2
  • Area = √(S(S - a)(S - b)(S - c))

Law of Tangents

• Less common formula: (a - b) / (a + b) = tan[(1/2)(A - B)] / tan[(1/2)(A + B)]

Conclusion

• Covered all key trigonometric identities and formulas needed for a typical trigonometry course or final exam.
• References to examples and practice problems available in the description section.