Introduction
What happens when you apply with ? You get . These "double-angle" formulas are incredibly useful shortcuts.
Prerequisite Connection
You know the Sum and Difference formulas for sine and cosine.
Today's Increment
We're deriving , (3 forms!), and .
Why This Matters
In Calculus, is solved by converting to a double-angle form. These formulas are essential for integration.
The Double-Angle Formulas
Sine Double-Angle
Tangent Double-Angle
Cosine Double-Angle (3 Forms)
Derivation: Start with , then use or to get the other two forms.
Worked Examples
Example 1: Evaluating a Double Angle
If and is in QI, find .
Find
Using
Since QI,
Apply Double-Angle Formula
Calculate
Answer:
Example 2: Simplifying an Expression
Simplify .
Recognize the Pattern
This is exactly the first form of !
Answer:
Example 3: Solving an Equation (Advanced)
Solve on .
Expand Double Angle
Rearrange and Factor
Solve Each Factor
Answer:
Common Pitfalls
Forgetting the "2" in front of
It's , not just .
Using the wrong cosine form
Choose the form that matches your given information. If you know sine, use .
Real-World Application
Physics: Projectile Motion
The range of a projectile is maximized when launched at 45°. The formula involves , which equals 1 when . This is why athletes throwing discus or javelin aim for that angle.
Practice Quiz
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