Unit 06 · Lesson 04 · Week 17 · Trigonometric Identities & Equations
Double-Angle and Half-Angle Formulas
Transform trigonometric expressions using reciprocal, quotient, Pythagorean, angle-sum, double-angle, and half-angle identities. In this lesson, the central focus is double-angle and half-angle formulas. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.
Learning intentions
- Interpret the defining quantities and notation used in double-angle and half-angle formulas.
- Construct or analyze at least two representations related to double-angle and half-angle formulas.
- Show symbolic work with correct domains, units, intervals, or angle measures.
- Explain why the selected method is valid and check the result against another representation.
Essential question
How can equivalent forms, identities, algebraic reasoning, and interval analysis simplify expressions and solve trigonometric equations?
Multiple-representation model
Transform trigonometric expressions using reciprocal, quotient, Pythagorean, angle-sum, double-angle, and half-angle identities.
\sin^2x+\cos^2x=1\tan x=\frac{\sin x}{\cos x}\sin(a+b)=\sin a\cos b+\cos a\sin b
Worked reasoning
Misconception check: To verify an identity, transform one side at a time into the other or into a common form; do not assume the statement is already true.
Communication and reasoning
Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.
Khaemenes Academy · Open-Age Precalculus
Double-Angle and Half-Angle Formulas
Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.
Name
Date
Rewrite tan x using sine and cosine.
Simplify 1−sin²x.
Simplify sec²x−tan²x.
Expand sin(a+b).
Write cos(2x) in terms of cosine only.
Evaluate sin75° exactly.