Double-Angle Identities Formula

Double-angle identities are formulas expressing (2), (2), and (2) in terms of single-angle trig functions.

The Formula

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta
cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta
tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

When to use: What if both angles in the sum formula are the same? Setting A=B=θA = B = \theta in the sum identities gives you the double-angle formulas. They answer: if you know the trig values for an angle, what are the trig values for twice that angle? The cosine double-angle formula is especially versatile because it has three equivalent forms, each useful in different situations—pick whichever one simplifies your problem.

Quick Example

sin⁡(2⋅30°)=2sin⁡30°cos⁡30°=2⋅12⋅32=32\sin(2 \cdot 30°) = 2\sin 30°\cos 30° = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}
Confirm: sin⁡60°=32\sin 60° = \frac{\sqrt{3}}{2}. \checkmark

Notation

The three forms of cos⁡(2θ)\cos(2\theta) are all equivalent. Use cos⁡2θ−sin⁡2θ\cos^2\theta - \sin^2\theta when you have both; 2cos⁡2θ−12\cos^2\theta - 1 when you only know cosine; 1−2sin⁡2θ1 - 2\sin^2\theta when you only know sine.

What This Formula Means

Formulas expressing sin⁡(2θ)\sin(2\theta), cos⁡(2θ)\cos(2\theta), and tan⁡(2θ)\tan(2\theta) in terms of single-angle trig functions.

What if both angles in the sum formula are the same? Setting A=B=θA = B = \theta in the sum identities gives you the double-angle formulas. They answer: if you know the trig values for an angle, what are the trig values for twice that angle? The cosine double-angle formula is especially versatile because it has three equivalent forms, each useful in different situations—pick whichever one simplifies your problem.

Formal View

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta; cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta; tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

Worked Examples

Example 1

easy
If cos⁡(θ)=35\cos(\theta) = \frac{3}{5}, find cos⁡(2θ)\cos(2\theta) using the double-angle formula.

Answer

cos⁡(2θ)=−725\cos(2\theta) = -\frac{7}{25}

First step

1
Use the double-angle formula: cos⁡(2θ)=2cos⁡2(θ)−1\cos(2\theta) = 2\cos^2(\theta) - 1.

Full solution

  1. 2
    Substitute: cos⁡(2θ)=2(35)2−1=2⋅925−1\cos(2\theta) = 2\left(\frac{3}{5}\right)^2 - 1 = 2 \cdot \frac{9}{25} - 1.
  2. 3
    =1825−1=1825−2525=−725= \frac{18}{25} - 1 = \frac{18}{25} - \frac{25}{25} = -\frac{7}{25}.
The double-angle formula for cosine has three equivalent forms: cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta. Choose the form that uses the information you have — here we used 2cos⁡2θ−12\cos^2\theta - 1 since we knew cosine.

Example 2

medium
Find sin⁡(2θ)\sin(2\theta) given that tan⁡(θ)=512\tan(\theta) = \frac{5}{12} and θ\theta is in Quadrant I.

Example 3

medium
Solve cos⁡(2x)=cos⁡x\cos(2x) = \cos x on [0,2π)[0, 2\pi).

Common Mistakes

  • Writing sin⁡2θ=2sin⁡θ\sin 2\theta=2\sin\theta - it is 2sin⁡θcos⁡θ2\sin\theta\cos\theta, the sum identity with equal angles.
  • Picking the wrong cos⁡2θ\cos 2\theta form - use 1−2sin⁡2θ1-2\sin^2\theta when you only know sine, 2cos⁡2θ−12\cos^2\theta-1 when you only know cosine.
  • Doubling the function for cosine too - cos⁡2θ≠2cos⁡θ\cos 2\theta\ne 2\cos\theta; it equals cos⁡2θ−sin⁡2θ\cos^2\theta-\sin^2\theta.

Why This Formula Matters

They are essential for power-reduction (rewriting cos⁡2θ\cos^2\theta to integrate it) and for solving equations that mix sin⁡θ\sin\theta with sin⁡2θ\sin 2\theta. The three forms of cos⁡2θ\cos 2\theta matter: picking the one in the variable you already have (sin⁡\sin or cos⁡\cos) is what makes a substitution collapse cleanly. Recognizing it by "Is the angle exactly twice another, so I can express it from single-angle trig values?" — rather than by familiar numbers — is what lets a student tell it apart from sum and difference identities and half-angle identities and pythagorean identity in a mixed problem set.

Frequently Asked Questions

What is the Double-Angle Identities formula?

Formulas expressing sin⁡(2θ)\sin(2\theta), cos⁡(2θ)\cos(2\theta), and tan⁡(2θ)\tan(2\theta) in terms of single-angle trig functions.

How do you use the Double-Angle Identities formula?

What if both angles in the sum formula are the same? Setting A=B=θA = B = \theta in the sum identities gives you the double-angle formulas. They answer: if you know the trig values for an angle, what are the trig values for twice that angle? The cosine double-angle formula is especially versatile because it has three equivalent forms, each useful in different situations—pick whichever one simplifies your problem.

What do the symbols mean in the Double-Angle Identities formula?

The three forms of cos⁡(2θ)\cos(2\theta) are all equivalent. Use cos⁡2θ−sin⁡2θ\cos^2\theta - \sin^2\theta when you have both; 2cos⁡2θ−12\cos^2\theta - 1 when you only know cosine; 1−2sin⁡2θ1 - 2\sin^2\theta when you only know sine.

Why is the Double-Angle Identities formula important in Math?

They are essential for power-reduction (rewriting cos⁡2θ\cos^2\theta to integrate it) and for solving equations that mix sin⁡θ\sin\theta with sin⁡2θ\sin 2\theta. The three forms of cos⁡2θ\cos 2\theta matter: picking the one in the variable you already have (sin⁡\sin or cos⁡\cos) is what makes a substitution collapse cleanly. Recognizing it by "Is the angle exactly twice another, so I can express it from single-angle trig values?" — rather than by familiar numbers — is what lets a student tell it apart from sum and difference identities and half-angle identities and pythagorean identity in a mixed problem set.

What do students get wrong about Double-Angle Identities?

The procedure for double-angle identities is the easy part; the trap is writing sin⁡2θ=2sin⁡θ\sin 2\theta=2\sin\theta. Asking "Is the angle exactly twice another, so I can express it from single-angle trig values?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Double-Angle Identities formula?

Before studying the Double-Angle Identities formula, you should understand: trig identities sum difference.

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