3.6.3 Double Angle Formulae

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

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

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

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

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