# 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.