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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.
例題 · polar form of 1 + i
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- 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).
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どうやって: 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).
よくある質問
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.
More in Precalculus
Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theorem