Worksheet: Euler's Formula for Trigonometric Identities

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

Q1:

Using Eulerโ€™s formula, express t a n ( 3 ๐œƒ ) in terms of t a n ๐œƒ .

  • A t a n t a n t a n t a n ( 3 ๐œƒ ) = 3 ๐œƒ โˆ’ ๐œƒ 1 โˆ’ 3 ๐œƒ ๏Šฉ ๏Šจ
  • B t a n t a n t a n t a n ( 3 ๐œƒ ) = โˆ’ 2 ๐œƒ + ๐œƒ 1 โˆ’ 2 ๐œƒ ๏Šฉ ๏Šจ
  • C t a n t a n t a n t a n ( 3 ๐œƒ ) = 2 ๐œƒ โˆ’ ๐œƒ 1 โˆ’ 2 ๐œƒ ๏Šฉ ๏Šจ
  • D t a n t a n t a n t a n ( 3 ๐œƒ ) = โˆ’ 3 ๐œƒ + ๐œƒ 1 โˆ’ 3 ๐œƒ ๏Šฉ ๏Šจ
  • E t a n t a n t a n t a n ( 3 ๐œƒ ) = 3 ๐œƒ + ๐œƒ 1 + 3 ๐œƒ ๏Šฉ ๏Šจ

Q2:

Using Eulerโ€™s formula, express s i n ๏Šซ ๐œƒ in the form ๐‘Ž ( 5 ๐œƒ ) + ๐‘ ( 3 ๐œƒ ) + ๐‘ ( ๐œƒ ) s i n s i n s i n , where ๐‘Ž , ๐‘ , and ๐‘ are constants to be found.

  • A s i n s i n s i n s i n ๏Šซ ๐œƒ = 1 1 6 ( ( 5 ๐œƒ ) + 1 0 ( 3 ๐œƒ ) โˆ’ 5 ( ๐œƒ ) )
  • B s i n s i n s i n s i n ๏Šซ ๐œƒ = 1 3 2 ( ( 5 ๐œƒ ) + 1 0 ( 3 ๐œƒ ) โˆ’ 5 ( ๐œƒ ) )
  • C s i n s i n s i n s i n ๏Šซ ๐œƒ = ( 5 ๐œƒ ) โˆ’ 5 ( 3 ๐œƒ ) + 1 0 ( ๐œƒ )
  • D s i n s i n s i n s i n ๏Šซ ๐œƒ = 1 1 6 ( ( 5 ๐œƒ ) โˆ’ 5 ( 3 ๐œƒ ) + 1 0 ( ๐œƒ ) )
  • E s i n s i n s i n s i n ๏Šซ ๐œƒ = 1 3 2 ( ( 5 ๐œƒ ) โˆ’ 5 ( 3 ๐œƒ ) + 1 0 ( ๐œƒ ) )

Q3:

Using Eulerโ€™s formula, derive a formula for c o s ( 3 ๐œƒ ) and s i n ( 3 ๐œƒ ) in terms of s i n ๐œƒ and c o s ๐œƒ .

  • A c o s c o s c o s ( 3 ๐œƒ ) = โˆ’ 2 ๐œƒ + 3 ๐œƒ ๏Šฉ , s i n s i n s i n ( 3 ๐œƒ ) = 3 ๐œƒ โˆ’ 2 ๐œƒ ๏Šฉ
  • B c o s c o s c o s ( 3 ๐œƒ ) = 4 ๐œƒ + 3 ๐œƒ ๏Šฉ , s i n s i n s i n ( 3 ๐œƒ ) = 3 ๐œƒ + 4 ๐œƒ ๏Šฉ
  • C c o s c o s c o s ( 3 ๐œƒ ) = 4 ๐œƒ โˆ’ 3 ๐œƒ ๏Šฉ , s i n s i n s i n ( 3 ๐œƒ ) = 3 ๐œƒ โˆ’ 4 ๐œƒ ๏Šฉ
  • D c o s c o s c o s s i n ( 3 ๐œƒ ) = ๐œƒ + 3 ๐œƒ ๐œƒ ๏Šฉ ๏Šจ , s i n s i n c o s s i n ( 3 ๐œƒ ) = โˆ’ 3 ๐œƒ ๐œƒ + ๐œƒ ๏Šจ ๏Šฉ
  • E c o s c o s c o s ( 3 ๐œƒ ) = โˆ’ 2 ๐œƒ โˆ’ 3 ๐œƒ ๏Šฉ , s i n s i n s i n ( 3 ๐œƒ ) = 3 ๐œƒ + 2 ๐œƒ ๏Šฉ

Q4:

Using Eulerโ€™s formula, express c o s ๏Šช ๐œƒ in the form ๐‘Ž ( 4 ๐œƒ ) + ๐‘ ( 2 ๐œƒ ) + ๐‘ c o s c o s , where ๐‘Ž , ๐‘ , and ๐‘ are constants to be found.

  • A c o s c o s c o s ๏Šช ๐œƒ = 1 1 6 ( 6 ( 4 ๐œƒ ) + 8 ( 2 ๐œƒ ) + 1 )
  • B c o s c o s c o s ๏Šช ๐œƒ = 1 1 6 ( 4 ( 4 ๐œƒ ) + ( 2 ๐œƒ ) + 6 )
  • C c o s c o s c o s ๏Šช ๐œƒ = 1 1 6 ( ( 4 ๐œƒ ) + 4 ( 2 ๐œƒ ) + 6 )
  • D c o s c o s c o s ๏Šช ๐œƒ = 1 8 ( ( 4 ๐œƒ ) + 4 ( 2 ๐œƒ ) + 3 )
  • E c o s c o s c o s ๏Šช ๐œƒ = 1 8 ( ( 4 ๐œƒ ) + 4 ( 2 ๐œƒ ) + 6 )

Q5:

Express c o s ๐œƒ and s i n ๐œƒ in terms of ๐‘’ ๏ƒ ๏ผ and ๐‘’ ๏Šฑ ๏ƒ ๏ผ .

  • A c o s ๐œƒ = 1 2 ๐‘– ๏€น ๐‘’ + ๐‘’ ๏… ๏ƒ ๏ผ ๏Šฑ ๏ƒ ๏ผ , s i n ๐œƒ = 1 2 ๏€น ๐‘’ โˆ’ ๐‘’ ๏… ๏ƒ ๏ผ ๏Šฑ ๏ƒ ๏ผ
  • B c o s ๐œƒ = 1 2 ๐‘’ ๏ƒ ๏ผ , s i n ๐œƒ = 1 2 ๐‘– ๐‘’ ๏Šฑ ๏ƒ ๏ผ
  • C c o s ๐œƒ = 1 2 ๏€น ๐‘’ + ๐‘’ ๏… ๏ƒ ๏ผ ๏Šฑ ๏ƒ ๏ผ , s i n ๐œƒ = 1 2 ๐‘– ๏€น ๐‘’ โˆ’ ๐‘’ ๏… ๏ƒ ๏ผ ๏Šฑ ๏ƒ ๏ผ
  • D c o s ๐œƒ = 1 2 ๐‘– ๏€น ๐‘’ โˆ’ ๐‘’ ๏… ๏ƒ ๏ผ ๏Šฑ ๏ƒ ๏ผ , s i n ๐œƒ = 1 2 ๏€น ๐‘’ + ๐‘’ ๏… ๏ƒ ๏ผ ๏Šฑ ๏ƒ ๏ผ
  • E c o s ๐œƒ = 1 2 ๏€น ๐‘’ โˆ’ ๐‘’ ๏… ๏ƒ ๏ผ ๏Šฑ ๏ƒ ๏ผ , s i n ๐œƒ = 1 2 ๐‘– ๏€น ๐‘’ + ๐‘’ ๏… ๏ƒ ๏ผ ๏Šฑ ๏ƒ ๏ผ

Q6:

Use Eulerโ€™s formula to express t a n ( 8 ๐œƒ ) in terms of t a n ๐œƒ .

Hint: First write c o s ( 8 ๐œƒ ) and s i n ( 8 ๐œƒ ) in terms of s i n ๐œƒ and c o s ๐œƒ .

  • A t a n t a n t a n t a n t a n t a n t a n t a n ( 8 ๐œƒ ) = 7 ๐œƒ โˆ’ 3 5 ๐œƒ + 2 1 ๐œƒ โˆ’ ๐œƒ 1 โˆ’ 2 1 ๐œƒ + 3 5 ๐œƒ โˆ’ 7 ๐œƒ ๏Šฉ ๏Šซ ๏Šญ ๏Šจ ๏Šช ๏Šฌ
  • B t a n t a n t a n t a n t a n t a n t a n t a n ( 8 ๐œƒ ) = โˆ’ 7 ๐œƒ + 3 5 ๐œƒ โˆ’ 2 1 ๐œƒ + ๐œƒ โˆ’ 1 + 2 1 ๐œƒ โˆ’ 3 5 ๐œƒ + 7 ๐œƒ ๏Šฉ ๏Šซ ๏Šญ ๏Šจ ๏Šช ๏Šฌ
  • C t a n t a n t a n t a n t a n t a n t a n t a n t a n ( 8 ๐œƒ ) = 8 ๐œƒ โˆ’ 5 6 ๐œƒ + 5 6 ๐œƒ โˆ’ 8 ๐œƒ 1 โˆ’ 2 8 ๐œƒ + 7 0 ๐œƒ โˆ’ 2 8 ๐œƒ + ๐œƒ ๏Šฉ ๏Šซ ๏Šญ ๏Šจ ๏Šช ๏Šฌ ๏Šฎ
  • D t a n t a n t a n t a n t a n t a n t a n t a n t a n ( 8 ๐œƒ ) = 8 ๐œƒ โˆ’ 5 6 ๐œƒ โˆ’ 5 6 ๐œƒ + 8 ๐œƒ 1 + 2 8 ๐œƒ โˆ’ 7 0 ๐œƒ + 2 8 ๐œƒ + ๐œƒ ๏Šฉ ๏Šซ ๏Šญ ๏Šจ ๏Šช ๏Šฌ ๏Šฎ
  • E t a n t a n t a n t a n t a n t a n t a n t a n t a n ( 8 ๐œƒ ) = โˆ’ 8 ๐œƒ + 5 6 ๐œƒ โˆ’ 5 6 ๐œƒ + 8 ๐œƒ 1 โˆ’ 2 8 ๐œƒ + 7 0 ๐œƒ โˆ’ 2 8 ๐œƒ + ๐œƒ ๏Šฉ ๏Šซ ๏Šญ ๏Šจ ๏Šช ๏Šฌ ๏Šฎ

Q7:

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

  • A c o s s i n ( 5 ๐œƒ ) โˆ’ ๐‘– ( 5 ๐œƒ )
  • B 5 ๐œƒ โˆ’ 5 ๐‘– ๐œƒ c o s s i n
  • C s i n c o s ( 5 ๐œƒ ) โˆ’ ๐‘– ( 5 ๐œƒ )
  • D c o s s i n ( 5 ๐œƒ ) + ๐‘– ( 5 ๐œƒ )
  • E 5 ๐œƒ + 5 ๐‘– ๐œƒ c o s s i n

Q8:

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

  • A c o s c o s s i n s i n c o s ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ + ๐œƒ ๐œ‘ , s i n c o s s i n s i n c o s ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ โˆ’ ๐œƒ ๐œ‘
  • B c o s c o s s i n s i n c o s ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ + ๐œƒ ๐œ‘ , s i n c o s c o s s i n s i n ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ โˆ’ ๐œƒ ๐œ‘
  • C c o s c o s c o s s i n s i n ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ + ๐œƒ ๐œ‘ , s i n s i n c o s c o s s i n ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ โˆ’ ๐œƒ ๐œ‘
  • D c o s c o s c o s s i n s i n ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ โˆ’ ๐œƒ ๐œ‘ , s i n c o s s i n s i n c o s ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ + ๐œƒ ๐œ‘
  • E c o s s i n c o s c o s s i n ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ โˆ’ ๐œƒ ๐œ‘ , s i n c o s c o s s i n s i n ( ๐œƒ โˆ’ ๐œ‘ ) = ๐œƒ ๐œ‘ + ๐œƒ ๐œ‘

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