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 for either or
- 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
- Substitute for either or
- To help solve trigonometric equations which contain tan 2θ
- Substitute tan 2θ for the double angle identity
- 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
Exam Tip
- 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 for . Show all working clearly.