Deriving and Applying Double and Half-Angle Identities

Miacademy & MiaPrep Learning Channel

This comprehensive math lesson guides students through the derivation and application of double-angle and half-angle identities in trigonometry. Building on prior knowledge of sum and difference identities, the video systematically proves formulas for sine, cosine, and tangent of double angles (e.g., 2θ) and half angles (e.g., θ/2). The narrator emphasizes understanding the logic behind these formulas rather than rote memorization, demonstrating how algebraic manipulation connects different trigonometric concepts. The video explores key themes such as algebraic verification of identities, the use of the Pythagorean identity to rewrite expressions, and the nuances of determining positive or negative signs based on quadrants. It specifically addresses the common student struggle of determining the correct sign for half-angle roots by analyzing quadrant locations. The lesson also contrasts general derivations with specific numerical applications, showing how to find exact values for non-standard angles like 22.5° and 112.5° without a calculator. For educators, this resource serves as both a theoretical lecture and a practical workshop. It is structured with built-in pause points that encourage active participation, allowing students to attempt derivations and problems before seeing the solution. The video serves as an excellent core lesson for Pre-Calculus or Trigonometry units, providing clear visual steps for complex algebraic substitutions and offering opportunities for error analysis and critical thinking regarding trigonometric properties.