Double-angle identities: what it means and how to use it
Rewrite trigonometric functions of twice an angle.
Double-angle identities is one of 5 angle formulas formulas in the trigonometry section of this library, and it is used at high school · ap level.
Why double-angle identities works
These are the sum formulas with B set equal to A. Once both angles match, the two mixed terms in the sine expansion become copies of each other and merge into 2 sin θ cos θ. In the cosine version the two terms stay different, leaving cos²θ − sin²θ, and swapping in the Pythagorean identity produces the other two familiar forms.
What each symbol means
$\theta$ is an angle.
Double-angle identities: when it holds
Equivalent cosine forms include $1-2\sin^2\theta$ and $2\cos^2\theta-1$.
When it stops applying
Running the identity backwards does not pin down the angle. Knowing cos 2θ = 0.5 tells you 2θ could be 60 or 300 degrees, so θ is 30 or 150 degrees, and only the quadrant of θ stated in the problem decides between them.
Double-angle identities: a worked example
If $\sin\theta=3/5$ and $\cos\theta=4/5$, then $\sin2\theta=24/25$.
The mistake to avoid
What people do: Students pull the 2 out front and write sin 2θ = 2 sin θ.
Why it goes wrong: The 2 multiplies the angle, not the function. At θ = 30 degrees the shortcut gives 1 while sin 60 degrees is 0.866, and at θ = 90 degrees it gives 2, which sine can never reach.
Do this instead: Use both factors: 2 sin 30 cos 30 = 2(0.5)(0.866) = 0.866. If sin θ = 3/5 and cos θ = 4/5, then sin 2θ = 2(3/5)(4/5) = 24/25.
Double-angle identities: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\theta$ is an angle.
- Check the conditions before substituting. Equivalent cosine forms include $1-2\sin^2\theta$ and $2\cos^2\theta-1$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Angle formulas
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where double-angle identities comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Trigonometric Identities lesson — the lesson behind this formula: simplify and prove relationships between trig functions.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about double-angle identities
Which of the three cosine forms should I pick?
Pick the one that matches what you already know. If you have only sine, use 1 − 2 sin²θ; if you have only cosine, use 2 cos²θ − 1; if you have both, the difference of squares is quickest.
Is there a double-angle rule for tangent as well?
Yes: tan 2θ = 2 tan θ / (1 − tan²θ). It is the tangent sum formula with both angles equal, and it goes undefined at θ = 45 degrees because the bottom hits zero there.
How do I handle sin 3θ or sin 4θ?
Build up in stages. Write 4θ as 2(2θ) and apply the rule twice, or write 3θ as 2θ + θ and use the sum formula, then expand the double angle inside.
Why is sin 2θ never bigger than 1 even with a 2 in front?
Because sin θ and cos θ are both fractions of size 1 or less, their product cannot exceed 1/2. That happens at 45 degrees, where 2(0.707)(0.707) is exactly 1.