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Double Angle Identities

Double Angle Identities
Double Number Identities

Trig identities that show how to find the sine, cosine, or tangent of twice a given angle.

 

Double Angle Identities

Math formula: sin 2x = 2 sin x cos x, the double angle identity for sine.

cos 2x equals three equivalent forms: cos²x − sin²x, 1 − 2sin²x, and 2cos²x − 1

tan(2x) = (2·tan x) / (1 − tan²x)

 

See also

Half angle identities

Key Formula

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta cos⁡(2θ)=2cos⁡2θ−1\cos(2\theta) = 2\cos^2\theta - 1 cos⁡(2θ)=1−2sin⁡2θ\cos(2\theta) = 1 - 2\sin^2\theta tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}
Where:
  • θ\theta = The original angle, measured in degrees or radians
  • 2θ2\theta = The double angle — twice the original angle

Worked Example

Problem: Given that θ is in the first quadrant and sin θ = 3/5, find sin(2θ) and cos(2θ).
Step 1: Find cos θ using the Pythagorean identity. Since θ is in the first quadrant, cos θ is positive.
cos⁡θ=1−sin⁡2θ=1−925=1625=45\cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}
Step 2: Apply the double angle identity for sine.
sin⁡(2θ)=2sin⁡θcos⁡θ=2⋅35⋅45=2425\sin(2\theta) = 2\sin\theta\cos\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}
Step 3: Apply the double angle identity for cosine (using the first form).
cos⁡(2θ)=cos⁡2θ−sin⁡2θ=1625−925=725\cos(2\theta) = \cos^2\theta - \sin^2\theta = \frac{16}{25} - \frac{9}{25} = \frac{7}{25}
Step 4: Verify using the Pythagorean identity on the results: sin²(2θ) + cos²(2θ) should equal 1.
(2425)2+(725)2=576+49625=625625=1  ✓\left(\frac{24}{25}\right)^2 + \left(\frac{7}{25}\right)^2 = \frac{576 + 49}{625} = \frac{625}{625} = 1 \;\checkmark
Answer: sin(2θ) = 24/25 and cos(2θ) = 7/25.

Another Example

This example shows how to use double angle identities in reverse — recognizing the pattern within a larger expression to simplify it, rather than expanding a double angle.

Problem: Simplify the expression 4 sin(15°) cos(15°) using a double angle identity.
Step 1: Recognize that the expression contains the pattern 2 sin θ cos θ. Factor out the 2 to match the identity.
4sin⁡(15°)cos⁡(15°)=2⋅2sin⁡(15°)cos⁡(15°)4\sin(15°)\cos(15°) = 2 \cdot 2\sin(15°)\cos(15°)
Step 2: Apply the double angle identity sin(2θ) = 2 sin θ cos θ with θ = 15°.
2⋅2sin⁡(15°)cos⁡(15°)=2sin⁡(2⋅15°)=2sin⁡(30°)2 \cdot 2\sin(15°)\cos(15°) = 2\sin(2 \cdot 15°) = 2\sin(30°)
Step 3: Evaluate sin(30°), which is a known exact value.
2sin⁡(30°)=2⋅12=12\sin(30°) = 2 \cdot \frac{1}{2} = 1
Answer: 4 sin(15°) cos(15°) = 1.

Frequently Asked Questions

Why are there three versions of the double angle formula for cosine?
All three forms are equivalent. They come from substituting the Pythagorean identity sin²θ + cos²θ = 1 into the first form cos(2θ) = cos²θ − sin²θ. Replacing sin²θ with 1 − cos²θ gives cos(2θ) = 2cos²θ − 1. Replacing cos²θ with 1 − sin²θ gives cos(2θ) = 1 − 2sin²θ. You choose whichever form best fits the information you already have.
How are double angle identities derived?
They come directly from the angle addition formulas. Start with sin(A + B) and set A = B = θ to get sin(2θ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ. Similarly, set A = B = θ in cos(A + B) = cos A cos B − sin A sin B to get cos(2θ) = cos²θ − sin²θ. The tangent double angle formula follows the same approach using the tangent addition formula.
When do you use double angle identities vs half angle identities?
Use double angle identities when you know the trig values of θ and need to find values of 2θ, or when simplifying expressions that contain products like sin θ cos θ. Use half angle identities when you know the trig values of θ and need to find values of θ/2. Half angle identities are actually rearranged forms of the double angle cosine formulas.

Double Angle Identities vs. Half Angle Identities

Double Angle IdentitiesHalf Angle Identities
PurposeFind trig values of 2θ from θFind trig values of θ/2 from θ
Sine formulasin(2θ) = 2 sin θ cos θsin(θ/2) = ±√[(1 − cos θ)/2]
Cosine formulacos(2θ) = cos²θ − sin²θcos(θ/2) = ±√[(1 + cos θ)/2]
Contains a ± sign?No — result is uniqueYes — sign depends on the quadrant of θ/2
RelationshipDerived from addition formulasDerived by solving double angle cosine formulas for sin θ or cos θ

Why It Matters

Double angle identities appear constantly in precalculus and calculus. In calculus, the identity cos(2θ) = 1 − 2sin²θ is rearranged to write sin²θ = (1 − cos 2θ)/2, which is essential for integrating powers of sine and cosine (power-reduction). These identities also simplify solving trigonometric equations and are used in physics to analyze projectile motion, where the range formula involves sin(2θ).

Common Mistakes

Mistake: Writing sin(2θ) = 2 sin θ, forgetting the cos θ factor.
Correction: The correct identity is sin(2θ) = 2 sin θ cos θ. Doubling an angle does not simply double the sine value. You must include both the sine and cosine of the original angle.
Mistake: Confusing cos(2θ) = cos²θ − sin²θ with cos(2θ) = cos²θ + sin²θ.
Correction: The sum cos²θ + sin²θ always equals 1 (Pythagorean identity) — it has nothing to do with the double angle. The double angle formula has a minus sign: cos²θ − sin²θ.

Related Terms

  • Trig Identities — The broader family of identities these belong to
  • Half Angle Identities — Derived from rearranging double angle formulas
  • Sine — One of the three functions with a double angle formula
  • Cosine — Has three equivalent double angle forms
  • Tangent — Has a double angle formula involving tan²θ
  • Angle — The geometric quantity being doubled
  • Pythagorean Identities — Used to derive alternate cosine double angle forms