Double Angle Formulae (DP IB Analysis & Approaches (AA): HL): Revision Note

Double angle formulae

What are the double angle formulas?

• The formulas for the double angle identities are:

  • $\sin 2\theta =2\sin \theta \cos \theta$

  •  $\begin{array}{rcl}\cos 2\theta & =& {\cos }^2\theta -\mathrm{ }{\sin }^2\theta ={2\cos }^2\theta -1 ={1-\mathrm{ }2\sin }^2\theta \end{array}$

  • $\tan 2\theta = \frac{2\tan \theta }{1-{\tan }^2\theta }$

How can I derive the double angle formulas?

• You can use the compound angle formulas to derive the double angle formulas

  • $\sin 2\theta =\sin \left(\theta +\theta \right)=\sin \theta \cos \theta +\cos \theta \sin \theta =2\sin \theta \cos \theta$

  • $\cos 2\theta =\cos \left(\theta +\theta \right)=\cos \theta \cos \theta -\sin \theta \sin \theta ={\cos }^2\theta -{\sin }^2\theta$

  • $\tan 2\theta =\tan \left(\theta +\theta \right)=\frac{\tan \theta +\tan \theta }{1-\tan \theta \tan \theta }=\frac{2\tan \theta }{1-{\tan }^2\theta }$

• You can then use ${\cos }^2\theta +{\sin }^2\theta =1$ to derive the other two versions of the cosine double angle formula

  • $\cos 2\theta ={\cos }^2\theta -\left(1-{\cos }^2\theta \right)=2{\cos }^2\theta -1$

  • $\cos 2\theta =\left(1-{\sin }^2\theta \right)-{\sin }^2\theta =1-2{\sin }^2\theta$

How can I rewrite expressions using the double angle formulas?

• The formulas can be used on a single term

  • For example, you can use the double angle formula on $\sin 6\theta$

  • The angle in the new expression is half of the original angle

    • $\sin 6\theta =2\sin 3\theta \cos 3\theta$

• The formulas can be used to write an expression as a single term

  • For example, you can use the double angle formula on ${\cos }^{2}5\theta -{\sin }^{2}5\theta$

  • The angle in the single term is double the original angle

    • ${\cos }^{2}5\theta -{\sin }^{2}5\theta =\cos 10\theta$

• You might need to factorise an expression or multiply an expression by a constant

  • For example, $2-4{\sin }^{2}3\theta =2\left(1-2{\sin }^{2}3\theta \right)=2\cos 6\theta$

  • For example, $4\sin 8\theta =4\left(2\sin 4\theta \cos 4\theta \right)=8\sin 4\theta \cos 4\theta$

How can I use the double angle formulas to solve equations?

• You want to try to write an equation in terms of only one trig function with the same angle

• If you have two (or more) expressions with different angles

  • Try using a double angle formula on the one with the bigger angle

    • $\cos 2\theta +3\sin \theta$ can be written as $1-2{\sin }^2\theta +3\sin \theta$

• If you have two (or more) different expressions with the same angle

  • Try factorising the expression

    • $4\sin \theta \cos \theta -3\cos \theta$ can be written as $\left(4\sin \theta -3\right)\cos \theta$