Lesson Worksheet: Euler’s Formula for Trigonometric Identities Mathematics

In this worksheet, we will practice using Euler’s formula to prove trigonometric identities like double angle and half angle.

Q1:

Use Eulerโ€™s formula to express tan(8๐œƒ) in terms of tan๐œƒ.

Hint: First write cos(8๐œƒ) and sin(8๐œƒ) in terms of sin๐œƒ and cos๐œƒ.

  • Atantantantantantantantan(8๐œƒ)=7๐œƒโˆ’35๐œƒ+21๐œƒโˆ’๐œƒ1โˆ’21๐œƒ+35๐œƒโˆ’7๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ
  • Btantantantantantantantantan(8๐œƒ)=8๐œƒโˆ’56๐œƒโˆ’56๐œƒ+8๐œƒ1+28๐œƒโˆ’70๐œƒ+28๐œƒ+๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ๏Šฎ
  • Ctantantantantantantantantan(8๐œƒ)=โˆ’8๐œƒ+56๐œƒโˆ’56๐œƒ+8๐œƒ1โˆ’28๐œƒ+70๐œƒโˆ’28๐œƒ+๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ๏Šฎ
  • Dtantantantantantantantan(8๐œƒ)=โˆ’7๐œƒ+35๐œƒโˆ’21๐œƒ+๐œƒโˆ’1+21๐œƒโˆ’35๐œƒ+7๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ
  • Etantantantantantantantantan(8๐œƒ)=8๐œƒโˆ’56๐œƒ+56๐œƒโˆ’8๐œƒ1โˆ’28๐œƒ+70๐œƒโˆ’28๐œƒ+๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ๏Šฎ

Q2:

Using Eulerโ€™s formula, express tan(3๐œƒ) in terms of tan๐œƒ.

  • Atantantantan(3๐œƒ)=โˆ’3๐œƒ+๐œƒ1โˆ’3๐œƒ๏Šฉ๏Šจ
  • Btantantantan(3๐œƒ)=3๐œƒ+๐œƒ1+3๐œƒ๏Šฉ๏Šจ
  • Ctantantantan(3๐œƒ)=โˆ’2๐œƒ+๐œƒ1โˆ’2๐œƒ๏Šฉ๏Šจ
  • Dtantantantan(3๐œƒ)=2๐œƒโˆ’๐œƒ1โˆ’2๐œƒ๏Šฉ๏Šจ
  • Etantantantan(3๐œƒ)=3๐œƒโˆ’๐œƒ1โˆ’3๐œƒ๏Šฉ๏Šจ

Q3:

Using Eulerโ€™s formula, derive a formula for cos(3๐œƒ) and sin(3๐œƒ) in terms of sin๐œƒ and cos๐œƒ.

  • Acoscoscos(3๐œƒ)=โˆ’2๐œƒโˆ’3๐œƒ๏Šฉ, sinsinsin(3๐œƒ)=3๐œƒ+2๐œƒ๏Šฉ
  • Bcoscoscossin(3๐œƒ)=๐œƒ+3๐œƒ๐œƒ๏Šฉ๏Šจ, sinsincossin(3๐œƒ)=โˆ’3๐œƒ๐œƒ+๐œƒ๏Šจ๏Šฉ
  • Ccoscoscos(3๐œƒ)=4๐œƒ+3๐œƒ๏Šฉ, sinsinsin(3๐œƒ)=3๐œƒ+4๐œƒ๏Šฉ
  • Dcoscoscos(3๐œƒ)=4๐œƒโˆ’3๐œƒ๏Šฉ, sinsinsin(3๐œƒ)=3๐œƒโˆ’4๐œƒ๏Šฉ
  • Ecoscoscos(3๐œƒ)=โˆ’2๐œƒ+3๐œƒ๏Šฉ, sinsinsin(3๐œƒ)=3๐œƒโˆ’2๐œƒ๏Šฉ

Q4:

Using Eulerโ€™s formula, express sin๏Šซ๐œƒ in the form ๐‘Ž(5๐œƒ)+๐‘(3๐œƒ)+๐‘(๐œƒ)sinsinsin, where ๐‘Ž, ๐‘, and ๐‘ are constants to be found.

  • Asinsinsinsin๏Šซ๐œƒ=116((5๐œƒ)โˆ’5(3๐œƒ)+10(๐œƒ))
  • Bsinsinsinsin๏Šซ๐œƒ=132((5๐œƒ)โˆ’5(3๐œƒ)+10(๐œƒ))
  • Csinsinsinsin๏Šซ๐œƒ=(5๐œƒ)โˆ’5(3๐œƒ)+10(๐œƒ)
  • Dsinsinsinsin๏Šซ๐œƒ=116((5๐œƒ)+10(3๐œƒ)โˆ’5(๐œƒ))
  • Esinsinsinsin๏Šซ๐œƒ=132((5๐œƒ)+10(3๐œƒ)โˆ’5(๐œƒ))

Q5:

Express cos๐œƒ and sin๐œƒ in terms of ๐‘’๏ƒ๏ผ and ๐‘’๏Šฑ๏ƒ๏ผ.

  • Acos๐œƒ=12๏€น๐‘’+๐‘’๏…๏ƒ๏ผ๏Šฑ๏ƒ๏ผ, sin๐œƒ=12๐‘–๏€น๐‘’โˆ’๐‘’๏…๏ƒ๏ผ๏Šฑ๏ƒ๏ผ
  • Bcos๐œƒ=12๏€น๐‘’โˆ’๐‘’๏…๏ƒ๏ผ๏Šฑ๏ƒ๏ผ, sin๐œƒ=12๐‘–๏€น๐‘’+๐‘’๏…๏ƒ๏ผ๏Šฑ๏ƒ๏ผ
  • Ccos๐œƒ=12๐‘’๏ƒ๏ผ, sin๐œƒ=12๐‘–๐‘’๏Šฑ๏ƒ๏ผ
  • Dcos๐œƒ=12๐‘–๏€น๐‘’โˆ’๐‘’๏…๏ƒ๏ผ๏Šฑ๏ƒ๏ผ, sin๐œƒ=12๏€น๐‘’+๐‘’๏…๏ƒ๏ผ๏Šฑ๏ƒ๏ผ
  • Ecos๐œƒ=12๐‘–๏€น๐‘’+๐‘’๏…๏ƒ๏ผ๏Šฑ๏ƒ๏ผ, sin๐œƒ=12๏€น๐‘’โˆ’๐‘’๏…๏ƒ๏ผ๏Šฑ๏ƒ๏ผ

Q6:

Using Eulerโ€™s formula, express cos๏Šช๐œƒ in the form ๐‘Ž(4๐œƒ)+๐‘(2๐œƒ)+๐‘coscos, where ๐‘Ž, ๐‘, and ๐‘ are constants to be found.

  • Acoscoscos๏Šช๐œƒ=116(4(4๐œƒ)+(2๐œƒ)+6)
  • Bcoscoscos๏Šช๐œƒ=18((4๐œƒ)+4(2๐œƒ)+6)
  • Ccoscoscos๏Šช๐œƒ=116(6(4๐œƒ)+8(2๐œƒ)+1)
  • Dcoscoscos๏Šช๐œƒ=116((4๐œƒ)+4(2๐œƒ)+6)
  • Ecoscoscos๏Šช๐œƒ=18((4๐œƒ)+4(2๐œƒ)+3)

Q7:

Using Eulerโ€™s formula, express ๐‘’๏Šซ๏ƒ๏ผ in terms of sine and cosine.

  • Acossin(5๐œƒ)โˆ’๐‘–(5๐œƒ)
  • B5๐œƒ+5๐‘–๐œƒcossin
  • Ccossin(5๐œƒ)+๐‘–(5๐œƒ)
  • Dsincos(5๐œƒ)โˆ’๐‘–(5๐œƒ)
  • E5๐œƒโˆ’5๐‘–๐œƒcossin

Q8:

What trigonometric identities can be derived by applying Eulerโ€™s identity to ๐‘’๏ƒ(๏ผ๏Šฑ๏ŽŠ)?

  • Acoscoscossinsin(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘โˆ’๐œƒ๐œ‘, sincossinsincos(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘+๐œƒ๐œ‘
  • Bcossincoscossin(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘โˆ’๐œƒ๐œ‘, sincoscossinsin(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘+๐œƒ๐œ‘
  • Ccoscoscossinsin(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘+๐œƒ๐œ‘, sinsincoscossin(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘โˆ’๐œƒ๐œ‘
  • Dcoscossinsincos(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘+๐œƒ๐œ‘, sincoscossinsin(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘โˆ’๐œƒ๐œ‘
  • Ecoscossinsincos(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘+๐œƒ๐œ‘, sincossinsincos(๐œƒโˆ’๐œ‘)=๐œƒ๐œ‘โˆ’๐œƒ๐œ‘

Q9:

Use Eulerโ€™s formula to express ๐‘’๏Šฑ๏ƒ๏ผ in terms of sine and cosine.

  • A๐‘’=๐œƒ+๐‘–๐œƒ๏Šฑ๏ƒ๏ผcossin
  • B๐‘’=๐œƒโˆ’๐‘–๐œƒ๏Šฑ๏ƒ๏ผsincos
  • C๐‘’=โˆ’๐œƒ+๐‘–๐œƒ๏Šฑ๏ƒ๏ผsincos
  • D๐‘’=โˆ’๐œƒโˆ’๐‘–๐œƒ๏Šฑ๏ƒ๏ผcossin
  • E๐‘’=๐œƒโˆ’๐‘–๐œƒ๏Šฑ๏ƒ๏ผcossin

Given that ๐‘’๐‘’=1๏ƒ๏ผ๏Šฑ๏ƒ๏ผ, what trigonometric identity can be derived by expanding the exponentials in terms of trigonometric functions?

  • Acossincos๏Šจ๏Šจ๏Šจ๐œƒโˆ’๐œƒโ‰ก๐œƒ
  • Bsincos๐œƒ+๐œƒโ‰ก1
  • Csincos๏Šจ๏Šจ๐œƒ+๐œƒโ‰ก1
  • Dsincos๏Šจ๏Šจ๐œƒ+๐œƒโ‰กโˆ’1
  • Ecossin๏Šจ๏Šจ๐œƒโˆ’๐œƒโ‰กโˆ’1

Q10:

Use Eulerโ€™s formula to derive a formula for cos6๐œƒ and sin6๐œƒ in terms of sin๐œƒ and cos๐œƒ.

  • Acoscossincossincossin6๐œƒ=6๐œƒ๐œƒ+20๐œƒ๐œƒโˆ’6๐œƒ๐œƒ๏Šซ๏Šฉ๏Šฉ๏Šซ, sincoscossincossinsin6๐œƒ=๐œƒ+15๐œƒ๐œƒโˆ’15๐œƒ๐œƒโˆ’๐œƒ๏Šฌ๏Šช๏Šจ๏Šจ๏Šช๏Šฌ
  • Bcoscoscossincossinsin6๐œƒ=๐œƒ+15๐œƒ๐œƒ+15๐œƒ๐œƒ+๐œƒ๏Šฌ๏Šช๏Šจ๏Šจ๏Šช๏Šฌ, sincossincossincossin6๐œƒ=6๐œƒ๐œƒ+20๐œƒ๐œƒ+6๐œƒ๐œƒ๏Šซ๏Šฉ๏Šฉ๏Šซ
  • Ccoscoscossincossinsin6๐œƒ=โˆ’๐œƒโˆ’15๐œƒ๐œƒโˆ’15๐œƒ๐œƒโˆ’๐œƒ๏Šฌ๏Šช๏Šจ๏Šจ๏Šช๏Šฌ, sincossincossincossin6๐œƒ=โˆ’6๐œƒ๐œƒโˆ’20๐œƒ๐œƒโˆ’6๐œƒ๐œƒ๏Šซ๏Šฉ๏Šฉ๏Šซ
  • Dcoscoscossincossinsin6๐œƒ=๐œƒ+15๐œƒ๐œƒโˆ’15๐œƒ๐œƒโˆ’๐œƒ๏Šฌ๏Šช๏Šจ๏Šจ๏Šช๏Šฌ, sincossincossincossin6๐œƒ=6๐œƒ๐œƒ+20๐œƒ๐œƒโˆ’6๐œƒ๐œƒ๏Šซ๏Šฉ๏Šฉ๏Šซ
  • Ecoscoscossincossinsin6๐œƒ=๐œƒโˆ’15๐œƒ๐œƒ+15๐œƒ๐œƒโˆ’๐œƒ๏Šฌ๏Šช๏Šจ๏Šจ๏Šช๏Šฌ, sincossincossincossin6๐œƒ=6๐œƒ๐œƒโˆ’20๐œƒ๐œƒ+6๐œƒ๐œƒ๏Šซ๏Šฉ๏Šฉ๏Šซ

Hence, express tan6๐œƒ in terms of tan๐œƒ.

  • Atantantantantantantan6๐œƒ=6๐œƒ+20๐œƒโˆ’6๐œƒ1+15๐œƒโˆ’15๐œƒโˆ’๐œƒ๏Šฉ๏Šซ๏Šจ๏Šช๏Šฌ
  • Btantantantantantantan6๐œƒ=1+15๐œƒโˆ’15๐œƒโˆ’๐œƒ6๐œƒ+20๐œƒโˆ’6๐œƒ๏Šจ๏Šช๏Šฌ๏Šฉ๏Šซ
  • Ctantantantantantantan6๐œƒ=6๐œƒโˆ’20๐œƒ+6๐œƒ1โˆ’15๐œƒ+15๐œƒโˆ’๐œƒ๏Šฉ๏Šซ๏Šจ๏Šช๏Šฌ
  • Dtantantantantantantan6๐œƒ=โˆ’6๐œƒโˆ’20๐œƒโˆ’6๐œƒโˆ’1โˆ’15๐œƒโˆ’15๐œƒโˆ’๐œƒ๏Šฉ๏Šซ๏Šจ๏Šช๏Šฌ
  • Etantantantantantantan6๐œƒ=6๐œƒ+20๐œƒ+6๐œƒ1+15๐œƒ+15๐œƒ+๐œƒ๏Šฉ๏Šซ๏Šจ๏Šช๏Šฌ

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