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

Double Angle Formulae

What are the double angle formulae?

• The double angle formulae for sine and cosine are:

  • 

  •  

  • 

• 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 :

  • Substitute  for

  • Solve for , finding all values in the range for 

    • The range will need adapting for 

  • Find the solutions for 

• To help solve trigonometric equations which contain and or 

  • Substitute  for  

  • Isolate all terms in 

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

• To help solve trigonometric equations which contain  and or 

  • Substitute either or  for

    • Choose the trigonometric ratio that is already in the equation

  • Isolate all terms in 

  • 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