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

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

Double angle formulae

What are the double angle formulas?

  • The formulas for the double angle identities are:

    • sin space 2 theta blank equals space 2 sin theta cos space theta

    •  table row cell cos space 2 theta blank end cell equals cell space cos squared space theta minus blank sin squared space theta space equals space 2 cos squared space theta minus 1 blank equals space 1 minus blank 2 sin squared space theta end cell end table

    • tan 2 theta blank equals blank fraction numerator 2 tan invisible function application theta over denominator 1 minus tan squared invisible function application theta blank end fraction blank

Examiner Tips and Tricks

These can be found in the formula booklet.

  • The formulas for sin and cos can be found in the SL section

  • The formula for tan can be found in the HL section

How can I derive the double angle formulas?

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

    • sin 2 theta space equals space sin open parentheses theta space plus space theta close parentheses space equals space sin theta cos theta space plus space cos theta sin theta space equals space 2 sin theta cos theta

    • cos 2 theta space equals space cos open parentheses theta space plus space theta close parentheses space equals space cos theta cos theta space minus space sin theta sin theta space equals space cos squared theta space minus space sin squared theta

    • tan 2 theta space equals space tan open parentheses theta space plus space theta close parentheses space equals space fraction numerator tan theta space plus space tan theta over denominator 1 space minus space tan theta tan theta end fraction space equals space fraction numerator 2 tan theta over denominator 1 space minus space tan squared theta end fraction

  • You can then use cos squared theta space plus space sin squared theta space equals space 1 to derive the other two versions of the cosine double angle formula

    • cos 2 theta space equals space cos squared theta space minus space open parentheses 1 minus space cos squared theta close parentheses space equals space 2 cos squared theta space minus space 1

    • cos 2 theta space equals space open parentheses 1 minus sin squared theta close parentheses space minus space sin squared theta space equals space 1 space minus space 2 sin squared 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 space equals space 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 squared 5 theta space minus space sin squared 5 theta

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

      • cos squared 5 theta space minus space sin squared 5 theta space equals space cos 10 theta

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

    • For example, 2 space minus space 4 sin squared 3 theta space equals space 2 open parentheses 1 space minus space 2 sin squared 3 theta close parentheses space equals space 2 cos 6 theta

    • For example, 4 sin 8 theta space equals space 4 open parentheses 2 sin 4 theta cos 4 theta close parentheses space equals space 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 space plus space 3 sin theta can be written as 1 space minus space 2 sin squared theta space plus space 3 sin theta

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

    • Try factorising the expression

      • 4 sin theta cos theta space minus space 3 cos theta can be written as open parentheses 4 sin theta minus 3 close parentheses cos theta

Examiner Tips and Tricks

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 space 2 theta equals sin space theta for 0 degree space less or equal than space theta space less or equal than space 360 degree. Show all working clearly.

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

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.