Step by Step
1
The sine double angle formula
sin(2θ) = 2·sin(θ)·cos(θ). There's only one form of this formula, unlike cosine's.
2
The three equivalent forms of cosine's double angle formula
cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. All three forms are mathematically equivalent (derived from each other using the Pythagorean identity) — choose whichever form is most convenient based on what information you already have.
3
When to use which cosine form
If you only know cos(θ), use 2cos²θ − 1. If you only know sin(θ), use 1 − 2sin²θ. If you know both, cos²θ − sin²θ works directly.
4
Where these come from
Both double angle formulas are special cases of the angle-sum formulas (sin(A+B) and cos(A+B)), simply setting A = B = θ.
Applied Walkthrough
1
Given sin(θ) = 3/5 and θ is in Quadrant I, find sin(2θ) and cos(2θ).
2
First find cos(θ) using the Pythagorean identity: cos(θ) = 4/5 (as established in the earlier lesson on this same triangle).
3
Find sin(2θ) = 2·sin(θ)·cos(θ) = 2·(3/5)·(4/5) = 24/25.
4
Find cos(2θ) using the form that only needs sin(θ): cos(2θ) = 1 − 2sin²θ = 1 − 2(9/25) = 1 − 18/25 = 7/25.
Exam Application
Exams test whether you know all three equivalent forms of the cosine double angle formula and can select the most convenient one, and whether you can correctly apply both formulas given only partial information (like just sin(θ) or just cos(θ)).
⚠ Common Trap
The most common trap is trying to use the cos²θ − sin²θ form when you only know one of the two values — instead, pick the alternate form (2cos²θ−1 or 1−2sin²θ) that only requires the value you actually have.
✓ Quick Self-Check
1. What is the double angle formula for sine?
sin(2θ) = 2·sin(θ)·cos(θ).
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2. Name the three equivalent forms of the double angle formula for cosine.
cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
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3. If sin(θ) = 3/5 and cos(θ) = 4/5, what is sin(2θ)?
24/25.
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4. If you only know sin(θ), which form of the cosine double angle formula should you use?
cos(2θ) = 1 − 2sin²θ.
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5. What are the double angle formulas special cases of?
The angle-sum formulas sin(A+B) and cos(A+B), with A = B = θ.
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