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2-1 (c) The trigonometric (polar) form of a complex number

2-1 (c) The trigonometric (polar) form of a complex number

Point 3 of 8 — Lesson 1: Complex Numbers in Trigonometric and Exponential Form and De Moivres TheoremQuestion bank: 0 questions (0 with pictures)

03 2-1 (c) The trigonometric (polar) form of a complex number

The form

  • If z=x+yi has modulus r and principal amplitude θ, then x=rcos⁡θ and y=rsin⁡θ.
  • The trigonometric form (polar form) is: z=r(cos⁡θ+isin⁡θ)
  • The conversion is done in three steps: find r, find the quadrant, find θ.

Useful special values

  • 1=cos⁡0+isin⁡0, −1=cos⁡π+isin⁡π.
  • i=cos⁡π2+isin⁡π2, −i=cos⁡(−π2)+isin⁡(−π2).
  • cos⁡θ−isin⁡θ=cos⁡(−θ)+isin⁡(−θ).
✍️ Worked examples

Example 2: Write z=2−23 i in trigonometric form.

Solution:

  • x=2, y=−23, so z lies in the fourth quadrant.
  • r=4+12=4.
  • θ=tan⁡−1(−232)=tan⁡−1(−3)=−π3.
  • z=4[cos⁡(−π3)+isin⁡(−π3)].

Numbers that are not in standard form

  • A form such as −8(cos⁡45∘+isin⁡45∘) is not a trigonometric form because the coefficient is negative.
  • Use reduction formulas to turn it into r(cos⁡θ+isin⁡θ) with r>0.
  • Reduction rules: sin⁡(π2+θ)=cos⁡θ,  cos⁡(π2+θ)=−sin⁡θ,  sin⁡(π+θ)=−sin⁡θ,  cos⁡(π+θ)=−cos⁡θ.
✍️ Worked examples

Example 4: Find the modulus and principal amplitude of z=−8(cos⁡45∘+isin⁡45∘).

Solution:

  • −cos⁡45∘=cos⁡225∘ and −sin⁡45∘=sin⁡225∘.
  • z=8(cos⁡225∘+isin⁡225∘).
  • 225∘ is not in ]−180∘,180∘], so subtract 360∘: 225∘−360∘=−135∘.
  • ∣z∣=8 and the principal amplitude is −135∘=−3π4.

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