Cartesian/polar forms | Year 12

01 / LEARN

The idea, step by step

Two descriptions of one point

For $z = x + i y$ , the real part is $x$ and the imaginary part is the real number $y$ . In the Argand plane they are horizontal and vertical coordinates. Polar form instead gives the distance $r$ from the origin and an angle $θ$ measured anticlockwise from the positive real axis.

Write $cis θ = cos θ + i sin θ$ . Then $z = r cis θ$ , where $r \geq 0$ . Converting back uses $x = r cos θ$ and $y = r sin θ$ . Angles are in radians unless degrees are explicitly stated.

z = x + i y = r (cos θ + i sin θ)

Find an angle, not just a tangent

For a nonzero number, $r = \sqrt{x^{2} + y^{2}}$ . The ratio $y / x$ gives a tangent when $x \neq 0$ , but tangent alone cannot distinguish opposite quadrants. Locate the point first, then choose the angle.

Zero has modulus zero but no defined argument: every direction collapses to the same point. A polar representation of zero does not assign it a unique direction.

A common trap

Using ${tan}^{- 1} (y / x)$ without checking the quadrant.

Instead: The inverse tangent usually returns an angle between $- π / 2$ and $π / 2$ ; adjust it to the actual point.

02 / WORKED EXAMPLES

See the thinking

Example 1

Convert a second-quadrant point

Write $- \sqrt{3} + i$ in polar form.

$r = \sqrt{3 + 1} = 2$ .
The point is in quadrant II; its reference angle is $π / 6$ .
Thus $θ = π - π / 6 = 5 π / 6$ .

2 cis (5 π / 6)

Example 2

Recover exact coordinates

Write $4 cis (- π / 3)$ in Cartesian form.

$x = 4 cos (- π / 3) = 2$ .
$y = 4 sin (- π / 3) = - 2 \sqrt{3}$ .
Combine the coordinates as $x + i y$ .

2 - 2 \sqrt{3} i

Check your understanding

Give a polar form of $- 3 i$ .

Show answer

$3 cis (- π / 2)$ .

The distance is 3 and the point is directly below the origin.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Convert $1 + i$ to polar form.
2 marks
Convert $- 2 - 2 i$ to polar form with $- π < θ \leq π$ .
2 marks
Convert $6 cis (π / 6)$ to Cartesian form.
2 marks
A signal is represented by $5 cis π$ . State its real and imaginary components.
2 marks
A student writes $- 1 + i = \sqrt{2} cis (- π / 4)$ . Explain and correct the error.
3 marks
Why is an argument of $z = 0$ undefined?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Convert $1 + i$ to polar form.
2 marks
$r = \sqrt{1 + 1} = \sqrt{2}$ and the equal positive coordinates give $θ = π / 4$ .
$\sqrt{2} cis (π / 4)$ .
Convert $- 2 - 2 i$ to polar form with $- π < θ \leq π$ .
2 marks
The modulus is $\sqrt{8} = 2 \sqrt{2}$ . Quadrant III has principal angle $- 3 π / 4$ .
$2 \sqrt{2} cis (- 3 π / 4)$ .
Convert $6 cis (π / 6)$ to Cartesian form.
2 marks
Multiply 6 by $cos (π / 6) = \sqrt{3} / 2$ and $sin (π / 6) = 1 / 2$ .
$3 \sqrt{3} + 3 i$ .
A signal is represented by $5 cis π$ . State its real and imaginary components.
2 marks
$5 cos π = - 5$ and $5 sin π = 0$ .
Real component $- 5$ ; imaginary component $0$ .
A student writes $- 1 + i = \sqrt{2} cis (- π / 4)$ . Explain and correct the error.
3 marks
The given angle produces a positive real part and a negative imaginary part, opposite to those required.
The angle is in the wrong quadrant; use $\sqrt{2} cis (3 π / 4)$ .
Why is an argument of $z = 0$ undefined?
2 marks
Every value of $θ$ gives $0 cis θ = 0$ , so no direction is distinguished.
Zero has no direction from the origin.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics complex numbers: Cartesian and polar representations, de Moivre, roots and roots of unity, Argand geometry and loci, complex factorisation and conjugate roots of real polynomials. Real-polynomial techniques are taught in the separate Real Polynomials chapter.

Official curriculum source ↗

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