Sum and Difference Formulas

Trigonometry · Axiom Academy

LESSON Sum and Difference Formulas Powerful trigonometric identities that break down complex angles into simpler components, revealing beautiful geometric patterns on the unit circle. These formulas allow us to evaluate trigonometric functions of angle sums (A + B) and differences (A - B) using the individual angles: 2. Geometric Derivation: cos(A - B) Let's see where these formulas come from using the unit circle. We'll derive cos(A - B) geometrically by placing two angles on the unit circle and using the distance formula. Place angle A at point P₁ = (cos A, sin A) on the unit circle Place angle B at point P₂ = (cos B, sin B) on the unit circle The angle between them is (A - B) Rotate both points by -B so P₂ lands on (1, 0) Now P₁ is at angle (A - B): point P₃ = (cos(A-B), sin(A-B)) Since rotation preserves distances: d(P₁, P₂) = d(P₃, (1,0)) Apply distance formula to both sides and simplify! 3. Patterns and Memory Devices These formulas are easier to remember when you notice the underlying patterns: Let's use these formulas to find the exact value of sin(75°) by writing it as sin(45° + 30°), where we know the exact values for both angles.

This is the written version of the interactive lesson above. See the full Trigonometry course.