Polar-Complex Conversion: Step-by-Step Guide
A complex number can be written in rectangular form, a + bi, or in polar form, r(cos(theta) + i sin(theta)). In polar form, r is the distance from the origin and theta is the angle from the positive x-axis. To convert, use r = the square root of (a^2 + b^2) and theta = arctan(b/a), then adjust theta to be in the correct quadrant.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Rectangular to polar: find r. Given z = a + bi, find r = the square root of (a^2 + b^2). This is the distance from the origin.\[r = \sqrt{a^{2}+b^{2}}\]
- Find the angle theta in the right quadrant. Find the reference angle arctan(|b| / |a|). Then place it in the correct quadrant using the signs of a and b. Many calculators have atan2(b, a), which does this for you.\[\theta = \operatorname{atan2}(b,a)\]
- Write the polar form. Write r(cos(theta) + i sin(theta)), or r e^(i theta).\[z = r(\cos\theta + i\sin\theta) = r e^{i\theta}\]
- Polar to rectangular. Given r and theta, find a = r cos(theta) and b = r sin(theta). Then write a + bi.\[a = r\cos\theta,\quad b = r\sin\theta\]
Worked example
Convert z = -1 + sqrt(3) i to polar form.
Step 1: a = -1 and b = root 3. Find r: r = the square root of ((-1)^2 + (root 3)^2) = the square root of (1 + 3) = the square root of 4 = 2.
Step 2: The reference angle is arctan(root 3 / 1) = pi/3 (60 degrees).
Step 3: a is negative and b is positive, so z is in Quadrant II. The angle is theta = pi - pi/3 = 2 pi / 3.
Step 4: The polar form is z = 2(cos(2 pi / 3) + i sin(2 pi / 3)).
Check: 2 cos(2 pi / 3) = 2 x (-1/2) = -1, and 2 sin(2 pi / 3) = 2 x (root 3 / 2) = root 3. That gives back -1 + root 3 i.
Tips and common mistakes
- Always figure out the quadrant. The arctan button only gives angles between -pi/2 and pi/2.
- Quadrant I: theta = arctan(b/a). Quadrant II: theta = pi - arctan(|b/a|). Quadrant III: theta = -pi + arctan(b/a). Quadrant IV: theta = -arctan(|b/a|).
- r is never negative. If you get a negative r, add pi to theta and make r positive.
- Use radians or degrees, whichever the problem asks for. Be consistent.
- The form r e^(i theta) makes multiplying and taking powers of complex numbers easy.
- If a = 0, theta is pi/2 when b > 0, and -pi/2 (or 3 pi/2) when b < 0.
- Memorize the sine and cosine of pi/6, pi/4, pi/3, and pi/2 so you do not need a calculator.