Notes for the Physics-Based Calculus workshop

1 Trigonometric Functions We recall the following identities for trigonometric functions. Theorem 1.1. For all x ∈ R, cos(−x) = cos(x) and sin(−x) = − sin(x). Theorem 1.2. For all α, β ∈ R, cos(α + β) = cos(α) cos(β)− sin(α) sin(β). Theorem 1.3. For all α, β ∈ R, sin(α+ β) = sin(α) cos(β) + cos(α) sin(β). Proof. For small positive angles (so α+ β is less than a right angle), the identities from Theorem 1.2 and 1.3 both follow from looking at (and thinking about) the following diagram.