Double Angle Formulae

What are the double angle formulae?

The double angle formulae for sine and cosine are:
- $\sin 2 θ = 2 \sin θ \cos θ$
- $\begin{array}{rcl} \cos 2 θ & = & \cos^{2} θ - \sin^{2} θ = {2 \cos}^{2} θ - 1 = {1 - 2 \sin}^{2} θ \end{array}$
- $\tan 2 θ \equiv \frac{2 \tan θ}{1 - \tan^{2} θ}$
These can be found in the formula booklet
- The formulae for sin and cos can be found in the SL section
- The formula for tan can be found in the HL section

How are the double angle formulae derived?

The double angle formulae can be derived from the compound angle formulae
Simply replace B for A in each of the formulae and simplify
For example
- Sin 2A = sin (A + A) = sinAcosA + sinAcosA = 2sinAcosA

How are the double angle formulae used?

Double angle formulae will often be used with…
- ... trigonometry exact values
- ... graphs of trigonometric functions
- ... relationships between trigonometric ratios
To help solve trigonometric equations which contain $\sin θ \cos θ$ :
- Substitute $\frac{1}{2} \sin 2 θ$ for $\sin θ \cos θ$
- Solve for $2 θ$ , finding all values in the range for $2 θ$
  - The range will need adapting for $2 θ$
- Find the solutions for $θ$
To help solve trigonometric equations which contain $\sin 2 θ$ and $\sin θ$ or $\cos θ$
- Substitute $2 \sin θ \cos θ$ for $\sin 2 θ$
- Isolate all terms in $θ$
- Factorise or use another identity to write the equation in a form which can be solved
To help solve trigonometric equations which contain $\cos 2 θ$ and $\sin θ$ or $\cos θ$
- Substitute either ${2 \cos}^{2} θ - 1$ or ${1 - 2 \sin}^{2} θ$ for $\cos 2 θ$
  - Choose the trigonometric ratio that is already in the equation
- Isolate all terms in $θ$
- Solve
  - The equation will most likely be in the form of a quadratic
To help solve trigonometric equations which contain tan 2θ
- Substitute the double angle identity for tan 2θ
- Rearrange, often this will lead to a quadratic equation in terms of tan θ
- Solve
Double angle formulae can be used in proving other trigonometric identities

Examiner Tip

All these formulae are in the Topic 3: Geometry and Trigonometry section of the formula booklet
If you are asked to show that one thing is identical (≡) to another, look at what parts are missing – for example, if sinθ has disappeared you may want to choose the equivalent expression for cos2θ that does not include sinθ

Worked example

Without using a calculator, solve the equation $\sin 2 θ = \sin θ$ for $0 ° \leq θ \leq 360 °$ . Show all working clearly.

aa-sl-3-6-2-double-angle-formulae-w

Double Angle Formulae (DP IB Maths: AA HL)

Revision Note

What are the double angle formulae?

How are the double angle formulae derived?

How are the double angle formulae used?

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

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1.2.1 Introduction to Logarithms

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