Applications of de Moivre's Theorem (Edexcel A Level Further Maths: Core Pure)

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 

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

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