Polar Form of a Complex Number
Conversion Formula for rectangular to polar x + yi = r(cos θ + i sin θ)

Example 1: convert 5 + 2i to polar form
Step 1: sketch a graph
Step 2: find r using the Pythagorean Theorem
r^{2} = x^{2} + y^{2}
r = $\sqrt{{x}^{2}+{y}^{2}}$
r = $\sqrt{{5}^{2}+{2}^{2}}=\sqrt{29}\approx 5.4$
r = $\sqrt{{x}^{2}+{y}^{2}}$
r = $\sqrt{{5}^{2}+{2}^{2}}=\sqrt{29}\approx 5.4$
Step 3: Using Trigonometry find θ
Recall that
tan θ =
$\frac{y}{x}$ therefore
tan θ =
$\frac{2}{5}$
to find θ θ = ${\mathrm{tan}}^{1}\left(\frac{2}{5}\right)\approx 21.8\xb0$
to find θ θ = ${\mathrm{tan}}^{1}\left(\frac{2}{5}\right)\approx 21.8\xb0$
Step 4: write in polar form using the conversion formula
5 + 2i = 5.4 (cos 21.8° + i sin 21.8°)
 Example 2: convert 5  2i to polar form
Step 1: sketch a graph
Step 2: find r using the Pythagorean Theorem
r^{2} = x^{2} + y^{2}
r = $\sqrt{{x}^{2}+{y}^{2}}$
r = $\sqrt{{5}^{2}+{(2)}^{2}}=\sqrt{29}\approx 5.4$
r = $\sqrt{{x}^{2}+{y}^{2}}$
r = $\sqrt{{5}^{2}+{(2)}^{2}}=\sqrt{29}\approx 5.4$
Step 3: Using Trigonometry find θ
Because the complex number is in Quadrant IV and θ is the angle from the positive horizontal axis to the vector: θ = 360°  β
Recall that tan β
=
$\frac{y}{x}$
therefore tan β
=
$\frac{2}{5}$
to find β β = ${\mathrm{tan}}^{1}\left(\frac{2}{5}\right)\approx 21.8\xb0$
θ = 360°  β θ = 360°  21.8° = 338.2°
to find β β = ${\mathrm{tan}}^{1}\left(\frac{2}{5}\right)\approx 21.8\xb0$
θ = 360°  β θ = 360°  21.8° = 338.2°
Step 4: write in polar form using the conversion formula
5  2i = 5.4 (cos 338.2° + i sin 338.2°)
To write a complex number in polar form: (1) draw a sketch labeling all parts (2) use the Pythagorean theorem to find the length of r (3) Find θ by using trigonometry and (4) use the formula to write in polar form.
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