Double-Angle Identities Formula

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

The Formula

sin⁡(2θ)=2sin⁡θcos⁡θ
cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ
tan⁡(2θ)=2tan⁡θ1−tan⁡2θ

When to use: What if both angles in the sum formula are the same? Setting A=B=θ 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
Confirm: sin⁡60°=32. \checkmark

Notation

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

What This Formula Means

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

What if both angles in the sum formula are the same? Setting A=B=θ 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; cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ; tan⁡(2θ)=2tan⁡θ1−tan⁡2θ

Worked Examples

Example 1

easy
If cos⁡(θ)=35, find cos⁡(2θ) using the double-angle formula.

Answer

cos⁡(2θ)=−725

First step

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

Full solution

  1. 2
    Substitute: cos⁡(2θ)=2(35)2−1=2⋅925−1.
  2. 3
    =1825−1=1825−2525=−725.
The double-angle formula for cosine has three equivalent forms: cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ. Choose the form that uses the information you have — here we used 2cos⁡2θ−1 since we knew cosine.

Example 2

medium
Find sin⁡(2θ) given that tan⁡(θ)=512 and θ is in Quadrant I.

Example 3

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

Common Mistakes

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

Why This Formula Matters

They are essential for power-reduction (rewriting cos⁡2θ to integrate it) and for solving equations that mix sin⁡θ with sin⁡2θ. The three forms of cos⁡2θ matter: picking the one in the variable you already have (sin⁡ or 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), cos⁡(2θ), and tan⁡(2θ) 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=θ 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) are all equivalent. Use cos⁡2θ−sin⁡2θ when you have both; 2cos⁡2θ−1 when you only know cosine; 1−2sin⁡2θ when you only know sine.

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

They are essential for power-reduction (rewriting cos⁡2θ to integrate it) and for solving equations that mix sin⁡θ with sin⁡2θ. The three forms of cos⁡2θ matter: picking the one in the variable you already have (sin⁡ or 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. 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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