Mathematics
Grade 12
15 min
Convert complex numbers between rectangular and polar form
Convert complex numbers between rectangular and polar form
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Convert a complex number from rectangular form (a + bi) to its equivalent polar form (r(cosθ + isinθ)).
Accurately calculate the modulus (r) and argument (θ) of a complex number, ensuring the argument is in the correct quadrant.
Convert a complex number from polar form to its equivalent rectangular form.
Represent complex numbers graphically on the complex plane (Argand diagram) in both rectangular and polar forms.
Apply knowledge of the unit circle and special triangles to find exact trigonometric values during conversions.
Identify and correct common errors, such as quadrant mistakes for the argument and calculator mode issues.
How can you give directions to a treasure? 🗺️ You could say 'go 3 blocks east and 4 blocks north', or you could say &...
2
Key Concepts & Vocabulary
TermDefinitionExample
Rectangular FormA complex number expressed as a sum of its real and imaginary parts, written as z = a + bi. It corresponds to the Cartesian coordinates (a, b) on the complex plane.The complex number z = 4 - 3i is in rectangular form, with a real part of 4 and an imaginary part of -3.
Polar FormA complex number expressed using its distance from the origin (modulus, r) and its angle relative to the positive real axis (argument, θ). It is written as z = r(cosθ + isinθ).The complex number z = 5(cos(π/3) + isin(π/3)) is in polar form.
Complex Plane (Argand Diagram)A two-dimensional coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number.The number z = -2 + 5i is plotted at the point (-2,...
3
Core Formulas
Conversion from Rectangular to Polar Form
Given z = a + bi, find r and θ using: 1. r = \sqrt{a^2 + b^2} 2. \theta = \arctan(\frac{b}{a}) (must be adjusted for the correct quadrant).
Use these formulas to find the modulus (r) and argument (θ) from the real part (a) and imaginary part (b). Always check the signs of 'a' and 'b' to determine the correct quadrant for θ.
Conversion from Polar to Rectangular Form
Given z = r(cosθ + isinθ), find a and b using: 1. a = r \cos(\theta) 2. b = r \sin(\theta). Then write as z = a + bi.
Use these formulas to find the real part (a) and imaginary part (b) by distributing the modulus (r) and evaluating the trigonometric functions for the given argument (θ).
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
A student converts z = -4 + 4i to polar form and gets the answer 4√2(cos(-π/4) + isin(-π/4)). What was the student's primary conceptual error?
A.The modulus was calculated incorrectly.
B.The argument was calculated for the wrong quadrant.
C.The calculator was in degree mode.
D.The parentheses were forgotten in the final form.
Challenging
A complex number z has a modulus of 6 and lies on the line y = x in the complex plane. What are the two possible representations of z in polar form using the principal argument?
A.6(cos(π/4) + isin(π/4)) and 6(cos(3π/4) + isin(3π/4))
B.6(cos(π/4) + isin(π/4)) and 6(cos(-3π/4) + isin(-3π/4))
C.6(cos(π/4) + isin(π/4)) and 6(cos(5π/4) + isin(5π/4))
D.6(cos(π/2) + isin(π/2)) and 6(cos(-π/2) + isin(-π/2))
Challenging
Let z = -2 + 2√3i. Its polar form is 4(cos(2π/3) + isin(2π/3)). Let w be a complex number with the same argument as z, but with a modulus that is the reciprocal of z's modulus. What is w in rectangular form?
A.w = -1/2 + (√3/2)i
B.w = -1/4 + (√3/4)i
C.w = -1/8 + (√3/8)i
D.w = -2 - 2√3i
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free