Compound Angle Formulae

Understanding Compound Angle Formulae

Definition and Importance of Compound Angles

  • Acknowledge what compound angles are and their significance in trigonometry.
  • Recognise the inherent blend of algebra and trigonometry in these formulae, leading to useful values of sine, cosine, and tangent.

Mastering the Basic Compound Angle Formulae

  • Familiarise with the basic compound angle formulae for sine, cosine and tangent: sin(A ± B), cos(A ± B), and tan(A ± B).
  • Demonstrate ability to derive these formulae if required, using core trigonometric properties and identities.

Applying Compound Angle Formulae

  • Gain proficiency in using these formulae to simplify trigonometric expressions or solve equations.
  • Infer that because these formulae involve two angles, they have wide-ranging applications in mathematics, including complex numbers and analytic geometry.

Range of Applications

Compound Angle Formulae in Trigonometric Identities

  • Comprehend the use of compound angle formulae in proof and derivation of further trigonometric identities.
  • Note the role they play in the proofs of double-angle and half-angle identities.

Use in Exact Trigonometric Values

  • Understand their application in derivation of exact trigonometric values for the angles 15°, 30°, 45°, 60° and 90°.
  • Be capable of obtaining exact values for other angles using these formulae.

Utility in Trigonometric Equations

  • Respect the utility of compound angle formulae in solving trigonometric equations.
  • Be familiar with techniques for tackling an equation which involves the sum or difference of two angles.

Real-world Applications

  • Appreciate the practical applications of compound angle formulae in physics and geometry.
  • Be aware that they often provide solutions or simplifications in problems involving circles, waves, oscillation, and vectors.

Connection with Complex Numbers and De Moivre’s Theorem

  • Notice the connection between compound angle formulae and the geometry and arithmetic of complex numbers.
  • Be versed in how they help in the proof and application of De Moivre’s Theorem.
  • Understand their application in finding the nth roots of complex numbers.

Use in Analytic Geometry

  • Understand the use of compound angle formulae in finding the equation of a line, circle, or parabola in the complex plane.
  • Appreciate their service in solving problems of rotation and translation in the plane.