TRIGONOMETRIC IDENTITIES

Trigonometric Identities

• Basic Identities:

  • sin(-θ) = -sin(θ)

  • cos(-θ) = cos(θ)

  • tan(-θ) = -tan(θ)

  • cosec(-θ) = -cosec(θ)

  • sec(-θ) = sec(θ)

  • cot(-θ) = -cot(θ)

Angle Relationships

• Conversion between Radians and Degrees:

  • Radian measure = π/180 × Degree measure

  • Degree measure = 180/π × Radian measure

• Trigonometric Values at Key Angles:

  • cos(0°) = 1; sin(0°) = 0

  • cos(π/2) = 0; sin(π/2) = 1

  • cos(π) = -1; sin(π) = 0

  • cos(3π/2) = 0; sin(3π/2) = -1

  • cos(2π) = 1; sin(2π) = 0

Fundamental Identities

• Pythagorean Identities:

  • sin²(x) + cos²(x) = 1

  • 1 + tan²(x) = sec²(x)

  • 1 + cot²(x) = cosec²(x)

Sum and Difference Formulas

• Cosine Formulas:

  • cos(x+y) = cos(x)cos(y) - sin(x)sin(y)

  • cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

  • cos(π/2 - x) = sin(x)

  • cos(π - x) = -cos(x)

• Sine Formulas:

  • sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

  • sin(x-y) = sin(x)cos(y) - cos(x)sin(y)

  • sin(π/2 - x) = cos(x)

• Special Angle Identities:

  • sin(2x) = 2sin(x)cos(x)

  • cos(2x) = cos²(x) - sin²(x)

  • cos(2x) = 2cos²(x) - 1

  • cos(2x) = 1 - 2sin²(x)

Multiple Angle Formulas

• Tangent Formulas:

  • tan(2x) = 2tan(x)/(1-tan²(x))

  • tan(3x) = (3tan(x) - tan³(x))/(1 - 3tan²(x))

• Cotangent Formulas:

  • cot(x+y) = (cot(x)cot(y) - 1)/(cot(x) + cot(y))

  • cot(x-y) = (cot(x)cot(y) + 1)/(cot(y) - cot(x))

Summation and Difference of Sines and Cosines

• Sine Summation/Difference Formulas:

  • sin(x+y) + sin(x-y) = 2sin(x)cos(y)

  • sin(x+y) - sin(x-y) = 2cos(x)sin(y)

• Cosine Summation/Difference Formulas:

  • cos(x+y) + cos(x-y) = 2cos(x)cos(y)

  • cos(x+y) - cos(x-y) = -2sin(x)sin(y)

Additional Co-Function Identities

• sin(c) - sin(d) = 2cos((c+d)/2)sin((c-d)/2)

• cos(c) - cos(d) = -2sin((c+d)/2)sin((c-d)/2)

Key Takeaway

• Understanding these identities is crucial for solving various trigonometric problems.