Edexcel A Level Further Maths: Core Pure:复习笔记1.2.3 Applications of de Moivre's Theorem

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

Exam Tip

• You can use the substitutions c = cosθ and s = sinθ to shorten your working as long as you clearly state them and change back at the end of the proof

Worked Example

Prove that

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 eiθ?

• 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 eiθ
  • 
  • 
• STEP 2
  Use the binomial expansion to expand or
  • Simplify ik to one of i, -1, -i or 1
• STEP 3
  Due to symmetry you can pair terms up of the form  and  
  • Write as
  • 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 2k term!
  • coskθ can always be written as an expression using only terms of the form cosnθ
  • sinkθ 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

Worked Example

Prove that

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 eikθ to make it a geometric series
  • For example:
    • 
    • 
• You can now use the formulae to find an expression for the sum
  • The series involving eikθ 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

Exam Tip

• Exam questions normally lead you through this process
• It is common for questions to let C equal the sum of the series with cos and let S equal the sum of the series with sin
• You can then write C + iS which makes the trig terms becomes eikθ

Worked Example

Prove that