Mathematics
Grade 12
15 min
Convert complex numbers from polar to rectangular form
Convert complex numbers from polar to rectangular form
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1
Introduction & Learning Objectives
Learning Objectives
Identify the modulus (r) and argument (θ) from a complex number given in polar form.
Recall and state the conversion formulas for the real part (a) and the imaginary part (b).
Apply the conversion formulas to transform a complex number from polar form, z = r(cos θ + i sin θ), to rectangular form, z = a + bi.
Accurately evaluate trigonometric functions for special angles given in both degrees and radians.
Verify the resulting rectangular coordinates by considering the quadrant of the original angle.
Express the final answer in the standard rectangular form a + bi.
How do electrical engineers analyze the flow of alternating current, which has both a magnitude and a phase shift? ⚡ They use the exact conversion you're about to learn!
This tutorial will...
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Key Concepts & Vocabulary
TermDefinitionExample
Complex Number (Rectangular Form)A number expressed in the form z = a + bi, where 'a' is the real part and 'b' is the imaginary part. It corresponds to the point (a, b) on the complex plane.z = 3 + 4i
Complex Number (Polar Form)A number expressed in the form z = r(cos θ + i sin θ), where 'r' is the distance from the origin (modulus) and 'θ' is the angle from the positive real axis (argument).z = 5(\cos(53.1°) + i \sin(53.1°))
Modulus (r)The magnitude or length of the vector representing a complex number in the complex plane. It is the distance from the origin to the point (a, b) and is always non-negative.For the number z = 2(\cos(\pi/4) + i \sin(\pi/4)), the modulus is r = 2.
Argument (θ)The angle, measured counterclockwise fr...
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Core Formulas
Formula for the Real Part (a)
a = r \cos(\theta)
Use this formula to find the horizontal component (real part) of the complex number. Multiply the modulus by the cosine of the argument.
Formula for the Imaginary Part (b)
b = r \sin(\theta)
Use this formula to find the vertical component (imaginary part) of the complex number. Multiply the modulus by the sine of the argument.
The Conversion Identity
z = r(\cos(\theta) + i \sin(\theta)) = (r \cos(\theta)) + i(r \sin(\theta)) = a + bi
This identity shows the complete relationship, directly substituting the formulas for 'a' and 'b' to form the rectangular representation.
4 more steps in this tutorial
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Challenging
Using the angle sum identity for sin(A+B) and cos(A+B), convert z = 4(cos(75°) + i sin(75°)) to rectangular form. (Hint: 75° = 45° + 30°)
A.√6 - √2 + (√6 + √2)i
B.√6 + √2 + (√6 - √2)i
C.√3 - 1 + (√3 + 1)i
D.√3 + 1 + (√3 - 1)i
Challenging
A student is converting z = 6(cos(π/3) + i sin(π/3)) using a calculator. They get the result z ≈ 5.99 + 0.104i. What is the most likely cause of this error?
A.The student forgot to distribute the modulus 6.
B.The calculator was in Degrees mode instead of Radians.
C.The student swapped the values for sine and cosine.
D.The student used π ≈ 3.14 instead of the π key.
Challenging
Given z = √2(cos(π/4) + i sin(π/4)), find the rectangular form of z².
A.2
B.1 + i
C.2i
D.1 - i
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