Lesson Worksheet: De Moivre’s Theorem for Trigonometric Identities Mathematics

In this worksheet, we will practice using de Moivre’s theorem to obtain trigonometric identities.

Q1:

Express sinοŠ©πœƒ in terms of the sines of multiples of πœƒ.

  • A12(3πœƒ+3πœƒ)sinsin
  • B14(3πœƒβˆ’3πœƒ)sinsin
  • C14(3πœƒ+3πœƒ)sinsin
  • D12(3πœƒβˆ’3πœƒ)sinsin

Q2:

Which of the following is equivalent to cos3πœƒ?

  • AcoscossinοŠ©οŠ¨πœƒ+3πœƒπœƒ
  • BcoscossinοŠ©οŠ¨πœƒβˆ’3πœƒπœƒ
  • CcoscossinοŠ©οŠ¨πœƒ+3πœƒπœƒ
  • D1βˆ’πœƒsin
  • EcoscossinοŠ©οŠ¨πœƒβˆ’3πœƒπœƒ

Q3:

Using de Moivre’s theorem, find the exact value of ο„Έπœƒπœƒο‘½οŽ‘οŠ¦οŠ©cosd.

  • Aβˆ’23
  • B0
  • C83
  • D23
  • E12

Q4:

Use de Moivre’s theorem to express sin4πœƒ in terms of power of sinπœƒ and cosπœƒ.

  • A4πœƒπœƒ+πœƒπœƒβˆ’4πœƒπœƒsincossincossincos
  • BsinsincossincosοŠͺοŠ©οŠ©πœƒ+2πœƒπœƒβˆ’4πœƒπœƒ
  • C4πœƒπœƒβˆ’4πœƒπœƒsincossincos
  • DcossincossincosοŠͺοŠ©οŠ©πœƒ+2πœƒπœƒβˆ’4πœƒπœƒ
  • E4πœƒπœƒβˆ’πœƒπœƒβˆ’4πœƒπœƒsincossincossincos

Q5:

Express cos6πœƒ in terms of powers of sinπœƒ and cosπœƒ.

  • AcoscossincossinsinοŠͺοŠͺοŠ¬πœƒ+15πœƒπœƒβˆ’15πœƒπœƒβˆ’πœƒ
  • BcoscossincossinsinοŠͺοŠͺοŠ¬πœƒβˆ’15πœƒπœƒ+15πœƒπœƒβˆ’πœƒ
  • C6πœƒπœƒ+20πœƒπœƒβˆ’6πœƒπœƒcossincossincossin
  • D6πœƒπœƒβˆ’20πœƒπœƒ+6πœƒπœƒcossincossincossin

Q6:

Express cosοŠ¬πœƒ in terms of cos6πœƒ, cos5πœƒ, cos4πœƒ, cos3πœƒ, cos2πœƒ, cosπœƒ, and any constant terms.

  • AcoscoscoscosοŠ¬πœƒ=6πœƒ+64πœƒ+152πœƒ+10
  • BcoscoscoscosοŠ¬πœƒ=1166πœƒ+384πœƒ+15162πœƒ+58
  • CcoscoscoscosοŠ¬πœƒ=26πœƒ+124πœƒ+302πœƒ+20
  • DcoscoscoscosοŠ¬πœƒ=1646πœƒ+3324πœƒ+15642πœƒ+532
  • EcoscoscoscosοŠ¬πœƒ=1326πœƒ+3164πœƒ+15322πœƒ+516

Q7:

Use de Moivre’s theorem to express sin5πœƒ in terms of powers of sinπœƒ.

  • Asinsinsinsin5πœƒ=βˆ’4πœƒβˆ’20πœƒ+5πœƒοŠ«οŠ©
  • Bsinsinsin5πœƒ=11πœƒβˆ’10πœƒοŠ«οŠ©
  • Csinsinsinsin5πœƒ=16πœƒβˆ’20πœƒ+5πœƒοŠ«οŠ©
  • Dsinsinsinsin5πœƒ=15πœƒβˆ’20πœƒ+5πœƒοŠ«οŠ©
  • Esinsinsin5πœƒ=16πœƒ+5πœƒοŠ«

By considering the solutions of sin5πœƒ=0, find an exact representation for sinοŠ¨ο€»πœ‹5.

  • AsinοŠ¨ο€»πœ‹5=5βˆ’βˆš58
  • BsinοŠ¨ο€»πœ‹5=1011
  • CsinοŠ¨ο€»πœ‹5=5+√58
  • DsinοŠ¨ο€»πœ‹5=βˆ’5+√302
  • EsinοŠ¨ο€»πœ‹5=13

Q8:

Using de Moivre’s theorem, find the exact value of ο„Έπœƒπœƒ.ο‘½οŽ‘οŠ¦οŠ­sind

  • Aβˆ’1635
  • B1,02435
  • C51235
  • D1635
  • Eβˆ’1,02435

Q9:

Express sin6πœƒ in terms of powers of sinπœƒ and cosπœƒ.

  • Asincossincossincossin6πœƒ=βˆ’6πœƒπœƒβˆ’20πœƒπœƒβˆ’6πœƒπœƒοŠ«οŠ©οŠ©οŠ«
  • Bsincossincossincossin6πœƒ=6πœƒπœƒβˆ’20πœƒπœƒ+6πœƒπœƒοŠ«οŠ©οŠ©οŠ«
  • Csincoscossincossinsin6πœƒ=πœƒβˆ’15πœƒπœƒ+15πœƒπœƒβˆ’πœƒοŠ¬οŠͺοŠͺ
  • Dsincossincossincossin6πœƒ=βˆ’6πœƒπœƒ+20πœƒπœƒβˆ’6πœƒπœƒοŠ«οŠ©οŠ©οŠ«
  • Esincossincossincossin6πœƒ=6πœƒπœƒ+20πœƒπœƒ+6πœƒπœƒοŠ«οŠ©οŠ©οŠ«

Express cos6πœƒ in terms of powers of sinπœƒ and cosπœƒ.

  • Acoscossincossincossin6πœƒ=6πœƒπœƒβˆ’20πœƒπœƒ+6πœƒπœƒοŠ«οŠ©οŠ©οŠ«
  • Bcoscoscossincossinsin6πœƒ=πœƒ+15πœƒπœƒ+15πœƒπœƒ+πœƒοŠ¬οŠͺοŠͺ
  • Ccoscoscossincossinsin6πœƒ=πœƒβˆ’15πœƒπœƒ+15πœƒπœƒ+πœƒοŠ¬οŠͺοŠͺ
  • Dcoscoscossincossinsin6πœƒ=πœƒβˆ’15πœƒπœƒ+15πœƒπœƒβˆ’πœƒοŠ¬οŠͺοŠͺ
  • Ecoscoscossincossinsin6πœƒ=πœƒβˆ’15πœƒπœƒβˆ’15πœƒπœƒβˆ’πœƒοŠ¬οŠͺοŠͺ

Hence, express tan6πœƒ in terms of powers of tanπœƒ.

  • Atantantantantantantan6πœƒ=6πœƒβˆ’20πœƒ+6πœƒ1βˆ’15πœƒ+15πœƒβˆ’πœƒοŠ©οŠ«οŠ¨οŠͺ
  • Btantantantantantantan6πœƒ=1βˆ’15πœƒ+15πœƒβˆ’πœƒ6πœƒβˆ’20πœƒ+6πœƒοŠ¨οŠͺ
  • Ctantantantantantantan6πœƒ=6πœƒ+20πœƒ+6πœƒ1+15πœƒ+15πœƒ+πœƒοŠ©οŠ«οŠ¨οŠͺ
  • Dtantantantantantantan6πœƒ=6πœƒβˆ’20πœƒ+6πœƒ1βˆ’15πœƒβˆ’15πœƒβˆ’πœƒοŠ©οŠ«οŠ¨οŠͺ
  • Etantantantantantantan6πœƒ=βˆ’6πœƒ+20πœƒβˆ’6πœƒ1βˆ’15πœƒβˆ’15πœƒβˆ’πœƒοŠ©οŠ«οŠ¨οŠͺ

Q10:

Express sincosοŠ¨οŠ©πœƒπœƒ in the form π‘Žπœƒ+𝑏3πœƒ+𝑐5πœƒcoscoscos, where π‘Ž,𝑏, and 𝑐 are constants to be found.

  • A2(2πœƒβˆ’5πœƒβˆ’3πœƒ)coscoscos
  • Bβˆ’116(2πœƒβˆ’5πœƒβˆ’3πœƒ)coscoscos
  • C116(2πœƒβˆ’5πœƒβˆ’3πœƒ)coscoscos
  • D2(βˆ’2πœƒ+5πœƒ+3πœƒ)coscoscos
  • E12(2πœƒβˆ’5πœƒβˆ’3πœƒ)coscoscos

Hence, find all the solutions of coscos5πœƒ+3πœƒ=0 in the interval 0β‰€πœƒ<πœ‹. Give your answers in exact form.

  • Aπœƒ=πœ‹8,3πœ‹8,πœ‹2
  • Bπœƒ=πœ‹8,3πœ‹8
  • Cπœƒ=πœ‹8,3πœ‹8,5πœ‹8,7πœ‹8
  • Dπœƒ=πœ‹8,3πœ‹8,πœ‹2,5πœ‹8,7πœ‹8
  • Eπœƒ=πœ‹2

This lesson includes 18 additional questions and 23 additional question variations for subscribers.

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