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

### 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

#### 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

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

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

Prove that