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

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Double Angle Formulae

What are the double angle formulae?

  • The double angle formulae for sine and cosine are:

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

    •  table attributes columnalign right center left columnspacing 0px end attributes row cell space 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

  • These can be found in the formula booklet

    • You do not need to know how to derive them

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 space theta cos space theta:

    • Substitute 1 half sin space 2 theta for sin space theta cos space theta space

    • Solve for 2 theta, finding all values in the range for 2 theta

      • The range will need adapting for 2 theta

    • Find the solutions for theta

  • To help solve trigonometric equations which contain sin space 2 theta and sin space theta or cos space theta

    • Substitute 2 sin space theta cos space theta for sin space 2 theta 

    • Isolate all terms in theta

    • Factorise or use another identity to write the equation in a form which can be solved

  • To help solve trigonometric equations which contain cos space 2 theta and sin space theta or cos space theta

    • Substitute either 2 cos squared space theta minus 1 blankor 1 minus 2 sin squared space theta for cos space 2 theta 

      • Choose the trigonometric ratio that is already in the equation

    • Isolate all terms in theta blank

    • Solve

      • The equation will most likely be in the form of a quadratic

  • Double angle formulae can be used in proving other trigonometric identities

Examiner Tips and Tricks

  • 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 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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    Amber

    Author: Amber

    Expertise: Maths

    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.