Modulus-argument form

The form r(cos θ + i sin θ)

If |z| = r and arg z = θ, the point is (r cos θ, r sin θ), so z = r cos θ + i r sin θ = r(cos θ + i sin θ).

The form needs r > 0 and a plus sign. 2(cos π/3 - i sin π/3) is not in this form: rewrite it as 2(cos(-π/3) + i sin(-π/3)), using cos(-θ) = cos θ and sin(-θ) = -sin θ.

Converting

• To a + bi: work out r cos θ and r sin θ. 4(cos π/6 + i sin π/6) = 2sqrt(3) + 2i.
• From a + bi: find r = |z| and θ = arg z from a sketch. -1 - i = sqrt(2)(cos(-3π/4) + i sin(-3π/4)).

Moduli of products and quotients

|z_1 z_2| = |z_1||z_2| and |z_1/z_2| = |z_1|/|z_2|. So |(3 + 4i)(5 - 12i)| = 5 × 13 = 65, without multiplying out.

It follows that |zⁿ| = |z|ⁿ. (The matching rules for arguments are not required in F1.)

Worked example

Express z = -1 - i in the form r(cos θ + i sin θ), with -π < θ ≤ π.

1. r = sqrt(1 + 1) = sqrt(2).
2. z is in the third quadrant, at angle π/4 below the negative real axis.
3. θ = -(π - π/4) = -3π/4.

Answer: z = sqrt(2)(cos(-3π/4) + i sin(-3π/4))

Check yourself

1. z = 4(cos π/6 + i sin π/6). Find the real part of z, to 3 decimal places.
   Show answer

   3.464. Re z = 4 cos(π/6) = 4 × sqrt(3)/2 = 2sqrt(3) = 3.464.

2. z = 4(cos π/6 + i sin π/6). Find the imaginary part of z.
   Show answer

   2. Im z = 4 sin(π/6) = 4 × 1/2 = 2.

3. Find |(3 + 4i)(5 - 12i)| without multiplying out.
   Show answer

   65. |3 + 4i| = 5 and |5 - 12i| = 13, so the modulus of the product is 5 × 13 = 65.

4. Find |(1 + i)⁶|.
   Show answer

   8. |1 + i| = sqrt(2), so |(1 + i)⁶| = (sqrt(2))⁶ = 8.

5. Find |(6 + 8i)/(1 - i)|, as a simplified surd.
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   5√2. |6 + 8i| = 10 and |1 - i| = sqrt(2), so the modulus is 10/sqrt(2) = 5sqrt(2).

6. Which is 3i in modulus-argument form?
   • 3(cos 0 + i sin 0)
   • 3(cos π + i sin π)
   • 3(cos(π/2) + i sin(π/2))
   • 3(cos(-π/2) + i sin(-π/2))
   Show answer

   3(cos(π/2) + i sin(π/2)). 3i is on the positive imaginary axis: r = 3 and θ = π/2.

7. Write 2(cos π/3 - i sin π/3) in the form r(cos θ + i sin θ), with -π < θ ≤ π. What is θ?
   • -π/3
   • π/3
   • 2π/3
   • -2π/3
   Show answer

   -π/3. cos π/3 - i sin π/3 = cos(-π/3) + i sin(-π/3), so θ = -π/3. The point 1 - sqrt(3)i is in the fourth quadrant.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP1
• Pearson Edexcel IAL Further Pure Mathematics 1 scheme of work

Last updated 8 November 2026. Spotted a mistake? Tell us.