Double-Angle Formulas
What happens when in the sum formulas? You get the double-angle formulas — compact expressions for , , and that appear everywhere in calculus and physics.
Introduction
The double-angle formulas are special cases of the sum formulas where both angles are the same. They're among the most frequently used identities in all of mathematics — from calculus integration to physics wave equations.
Past Knowledge
Sum formulas for sin, cos, tan (5.10–5.12) and Pythagorean identity.
Today's Goal
Derive and apply all double-angle formulas, including the three forms of .
Future Success
Half-angle formulas (5.14) are derived from the cosine double-angle forms.
Key Concepts
Derivation
Set in each sum formula:
, which simplifies to:
The Double-Angle Formulas
| Function | Formula |
|---|---|
Why Three Forms for Cosine?
Starting from , use the Pythagorean identity to replace either term:
- Replace →
- Replace →
Which Form to Use?
Choose the form that matches the information you have: use if you know cosine, if you know sine, and if you know both.
Worked Examples
Finding sin 2θ
Question: If and is in QI, find .
Step 1: Find (Pythagorean identity, QI → positive).
Step 2: Apply the formula:
Final Answer:
Finding cos 2θ (Choosing the Right Form)
Question: If , find .
Step 1: We know cosine, so use the form :
Final Answer:
Simplifying with Double-Angle
Question: Simplify .
Step 1: Recognize the pattern with :
Final Answer:
Common Pitfalls
Writing
The correct formula is , not . You cannot distribute the 2 inside the function — sine is not linear.
Forgetting the Three Cosine Forms
There are three equivalent forms for . Using the wrong one makes problems needlessly difficult. Pick the form that matches the given information.
Real-Life Applications
Power Reduction in Electrical Engineering
In AC power analysis, the instantaneous power in a circuit involves products like . The double-angle formula converts this to , which is much easier to average over a full cycle — directly yielding the RMS (root mean square) power that determines your electricity bill.
Practice Quiz
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