Converting 3 + 4i to Polar Form

Trigonometry · Axiom Academy

EXAMPLE Converting Complex Numbers to Polar Form Learn to convert from rectangular form (a + bi) to polar form (r∠θ) through systematic calculation of modulus and argument. Excellent work! You've successfully converted a complex number from rectangular to polar form. Here's what we learned: Modulus (r): The distance from the origin to the point in the complex plane, calculated using r = √(a² + b²). This gives us the magnitude of the complex number. Argument (θ): The angle from the positive real axis, calculated using θ = arctan(b/a). This gives us the direction of the complex number. Polar Form: Can be expressed as r∠θ, r(cos θ + i sin θ), or re^(iθ). All three forms are equivalent and useful in different contexts. Quadrant Awareness: Since 3 + 4i is in Quadrant I (both components positive), our angle θ = arctan(4/3) ≈ 53.13° is already correct. Be careful with other quadrants! Polar form is especially useful for multiplication and division of complex numbers, as well as finding powers and roots using De Moivre's Theorem!

This is the written version of the interactive lesson above. See the full Trigonometry course.