Proving the Double and Half Angle Formulas for Trigonometry (Precalculus - Trigonometry 27)

Overview

Professor Leonard demonstrates the derivation of double and half-angle trigonometric formulas, starting with the sum and difference identities for sine and cosine. He then uses the Pythagorean identity to generate three distinct formulas for cosine's double angle and derives two additional identities for sine-squared and cosine-squared. Finally, he shows how to derive the half-angle formulas for sine, cosine, and tangent by substituting theta with alpha/2 into the previously derived identities.

Key takeaways

Double angle formulas are derived by setting the two angles in the sum formulas (e.g., sin(α + β)) equal to each other (θ + θ).
The three distinct double angle formulas for cosine (cos² θ - sin² θ, 2 cos² θ - 1, 1 - 2 sin² θ) arise from manipulating the primary formula using the Pythagorean identity.
The identities for sin² θ and cos² θ (power-reducing formulas) are derived by algebraically solving the cosine double angle formulas for sin² θ and cos² θ.
Half-angle formulas are derived by substituting θ = α/2 into the power-reducing identities, effectively transforming formulas involving 2θ into formulas involving α.
The sign (positive or negative) for the half-angle formulas is determined by the specific quadrant in which the angle α/2 lies.

Chapters

0:00 Deriving the Double Angle Formula for Sine

Starts with the sum formula for sine: sin(α + β) = sin α cos β + cos α sin β.
Sets α = β = θ to derive sin(2θ) = 2 sin θ cos θ.
Highlights this as the sole double angle formula for sine, crucial for Calculus II.

5:19 Deriving and Manipulating Double Angle Formulas for Cosine

Starts with the sum formula for cosine: cos(α + β) = cos α cos β - sin α sin β.
Sets α = β = θ to derive the first cosine double angle formula: cos(2θ) = cos² θ - sin² θ.
Uses the Pythagorean identity (sin² θ + cos² θ = 1) to derive two additional formulas: cos(2θ) = 2 cos² θ - 1 and cos(2θ) = 1 - 2 sin² θ.

15:50 Deriving Identities for Sine-Squared and Cosine-Squared from Double Angle Formulas

Rearranges cos(2θ) = 2 cos² θ - 1 to solve for cos² θ: cos² θ = (1 + cos 2θ) / 2.
Rearranges cos(2θ) = 1 - 2 sin² θ to solve for sin² θ: sin² θ = (1 - cos 2θ) / 2.
These are often called power-reducing formulas.

20:31 Deriving the Double Angle Formula for Tangent and Tangent-Squared Identities

Starts with the sum formula for tangent: tan(α + β) = (tan α + tan β) / (1 - tan α tan β).
Sets α = β = θ to derive tan(2θ) = 2 tan θ / (1 - tan² θ).
Uses tan² θ = sin² θ / cos² θ and the derived power-reducing formulas to show tan² θ = (1 - cos 2θ) / (1 + cos 2θ).

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Leonard.