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Parametric equations and polar coordinates

Curves traced by a parameter, and the (r, θ) description of the plane.

A parametric curve gives x and y each as functions of t; polar coordinates locate a point by distance r and angle θ. Picture it: the same point (1, 1) is √2 at 45°; the unit circle is simply r = 1. Think it: complex numbers in exponential form are polar coordinates with multiplication built in.

Àwọn ààtò àwọn ìṣàmúlò-ètò · polar form of 1 + i

Precalculus · Complex-Numbers

Polar form of 1 + I

1 + i

Àwọn ìṣàmúlò-ètò

z = 1 + i = 1 + (1)i
Read off the real and imaginary parts.
|z| = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.4142
The modulus is the distance from the origin (Pythagoras).
\theta = \arg z = \frac{\pi}{4} \approx 0.78540
The argument is the angle from the positive real axis (watch the quadrant).
z = \sqrt{2}\left(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right) = \sqrt{2} e^{i \frac{\pi}{4}}
Polar and exponential forms (Euler).

Fi àwọn àgbèwọlé hàn

\sqrt{2} e^{i \frac{\pi}{4}}

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Bii o ṣe le: Parametric equations and polar coordinates

Read off the real and imaginary parts.
The modulus is the distance from the origin (Pythagoras).
The argument is the angle from the positive real axis (watch the quadrant).
Polar and exponential forms (Euler).

Àwọn Àtòjọ-ẹ̀yàn

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Diẹ̀ nínú Precalculus

Complex numbers Polynomial functions Rational functions Sequences and series The binomial theorem Conic sections Vectors Exponential and logarithmic functions Polynomial division and the remainder theorem