Introduction
The Double-Angle formulas give in terms of . By solving those formulas backwards, we can find expressions for and .
Prerequisite Connection
You know the Double-Angle formulas, especially the three forms of .
Today's Increment
We're deriving , , and .
Why This Matters
In Calculus II, integrals like are solved using the "power-reducing" version of these formulas.
The Half-Angle Formulas
Sine Half-Angle
Cosine Half-Angle
Tangent Half-Angle (Two Forms)
Important: The ± in front of the square root is determined by the quadrant of , NOT .
Worked Examples
Example 1: Finding Exact Values
Find the exact value of .
Identify the Relationship
Apply Cosine Half-Angle
(Positive since 22.5° is in QI)
Substitute and Simplify
Answer:
Example 2: Determining the Sign
If , find .
Find
, which is in Quadrant II.
Determine Sign
Sine is POSITIVE in QII.
Apply Formula
Answer:
Example 3: Using Tangent Form (Advanced)
Simplify .
Recognize the Pattern
This matches one of the tangent half-angle forms.
Answer:
Common Pitfalls
Using the quadrant of instead of
If (QIII), then is in QII. The sign depends on where lands!
Forgetting to simplify nested radicals
Always look for opportunities to simplify to .
Real-World Application
Optics & Lens Design
The refraction of light through lenses involves half-angle calculations. When designing camera lenses or telescope mirrors, optical engineers use these formulas to calculate how light bends at curved surfaces.
Practice Quiz
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