Applications of de Moivre's Theorem (Edexcel A Level Further Maths): Revision Note

Multiple Angle Formulae

de Moivre’s theorem can be applied to prove identities such as . This allows expressions involving multiple angles to be written as a polynomial function of a single trig function which makes it easier to solve equations involving different angles.

How do I write sinkθ or coskθ in terms of powers of sinθ or cosθ?

• STEP 1 Use de Moivre’s theorem to write  

• STEP 2 Use the binomial expansion to expand 

• STEP 3 Use  to simplify the expansion and group the real terms and the imaginary terms separately

• STEP 4 Equate the real parts of the expansion to cos kθ and equate the imaginary parts to sinkθ

• STEP 5 (Depending on the question) Use  to write the identity in terms of sinθ only or cosθ depending on what the question asks

  • coskθ  can always be written as a function of just cosθ

  • sinkθ can be written as a function of just sinθ when k is odd

    • When k is even sinkθ will be a function of sinθ multiplied by a factor of cosθ

How do I write tankθ in terms of powers of tanθ?

• STEP 1 Find expressions for sinkθ and coskθ using the previous method

• STEP 2 Use the identity 

• STEP 3 Divide each term in the fraction by the highest power of cosθ to write each term using powers of tanθ and secθ

• STEP 4 (Depending on the question) Write everything in terms of tanθ using the identity 

Powers of Trig Functions

de Moivre’s theorem can be applied to prove identities such as . This allows powers of a trig function to be written in terms of multiple angles which makes them easier to integrate.

How can I write coskθ and sinkθ in terms of e^iθ?

• Recall and by de Moivre’s theorem 

• It follows that 

• You can derive expressions for sinkθ and coskθ using:

  • 

    • 

  • 

    • 

How do I write powers of sinθ or cosθ in terms of sinkθ or coskθ?

• STEP 1 Write the trig term in terms of e^iθ

  • 

  • 

• STEP 2 Use the binomial expansion to expand or 

  • Simplify i^k to one of i, -1, -i or 1

• STEP 3 Due to symmetry you can pair terms up of the form and 

  • Write as or 

  • If k is even then there will be a term by itself as 

• STEP 4 Rewrite each pair in terms of cosnθ  or sinnθ

  •  

  • 

• STEP 5 Simplify the expression – remember the 2^k term!

  • cos^kθ can always be written as an expression using only terms of the form cosnθ

  • sin^kθ can be written as an expression using only terms of the form:

    • sinnθ if k is odd

    • cosnθ if k is even

How do I write powers of tanθ in terms of sinkθ or coskθ?

• Use  and use the previous steps

• Note that the expression will be in terms of multiple angles of sin & cos and not tan

Trig Series

de Moivre’s theorem can be applied to find formulae for the sum of trigonometric series such as 

How can I find the sum of geometric series involving complex numbers?

• The geometric series formulae work with complex numbers

  • and  (provided )

• Suppose w and z are two complex numbers then:

  • 

  • provided |z|<1

    • Compare these to the geometric series formulae with a=w and r=z

How can I find the sum of geometric series involving sinθ or cosθ?

• Using de Moivre’s theorem: 

• You can find coskθ and sinkθ by taking real and imaginary parts

  • 

  • 

• Rewrite the series using e^ikθ to make it a geometric series

  • For example:

    • 

    • 

• You can now use the formulae to find an expression for the sum

  • The series involving e^ikθ will be geometric so determine

    • whether it is finite or infinite

    • what is the value of a (the first term) and r (the common ratio)

• Once you have used the formula the denominator will be of the form

  • Multiply the numerator and denominator by  

  • The denominator will become real 

    • This is because 

• If your series involved sin terms then take the imaginary part of the sum

• If your series involved cos terms then take the real part of the sum