Identities in Trigonometry

Fundamental Trigonometric Identities

  • Understand the basic trigonometric identities of sine, cosine, and tangent.
  • Reciprocal identities: Understand that cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent, respectively.
  • Quotient identities: Recognise that tangent is the quotient of sine by cosine and cotangent is the reciprocal of tangent.
  • Pythagorean identities: Recognise and use the identity sin²θ + cos²θ = 1, and its derived identities sec²θ = 1 + tan²θ and csc²θ = 1 + cot²θ.

Co-Function Identities

  • Appreciate that sine, cosine, and tangent at (90 - θ) are equal to cosine, sine, and cotangent at θ, respectively.
  • Relate the secant, cosecant and cotangent at (90 - θ) to the cosecant, secant, and tangent at θ.

Even-Odd Identities

  • Understand that sine and cosecant are odd functions, whilst cosine, secant, tangent, and cotangent are even functions.
  • Be aware of the implications of these identities, particularly that sin(-θ) = -sinθ, cos(-θ) = cosθ, and tan(-θ) = -tanθ.

Double Angle Formulas

  • Apply identities for sine, cosine, and tangent of double angles to simplify expressions.
  • Be familiar with the expressions 2sinθcosθ, cos²θ - sin²θ, and 2tanθ/(1 - tan²θ) for sin2θ, cos2θ, and tan2θ, respectively.

Half-Angle Formulas

  • Use the half-angle formulas to simplify trigonometric expressions containing half-angles.
  • Recall that √[(1±cosθ)/2] and ± √[(1 - cosθ)/2] are the expressions for sin(θ/2) and cos(θ/2), respectively.

Sum and Difference Formulas

  • Apply the sum and difference formulas for sin, cos, and tan to evaluate trigonometric expressions.
  • Practice finding exact trigonometric function values for nonstandard angles using these formulas.

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

  • Expand the product of sine or cosine of two angles using the product-to-sum formulas and contract a sum or difference of two sines or cosines into a product using the sum-to-product formulas.
  • Be comfortable switching between these forms of trigonometric identities to make calculations and simplifications easier.