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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.

कार्यरत उदाहरण · polar form of 1 + i

Precalculus · Complex-Numbers

Polar form of 1 + I

1 + i

चरणानुक्रमे

  1. z = 1 + i = 1 + (1)i

    Read off the real and imaginary parts.

  2. |z| = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.4142

    The modulus is the distance from the origin (Pythagoras).

  3. \theta = \arg z = \frac{\pi}{4} \approx 0.78540

    The argument is the angle from the positive real axis (watch the quadrant).

  4. 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).

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

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मोफत खाते प्रत्येक शिकवणी वर टिपण्णी जोडते, तुम्ही पूर्ण केलेल्या गोष्टीचा नोंदवही, तुमच्या सोडविलेल्या समस्या एका ठिकाणी, आणि या पानाच्या विषयी तुम्ही विचारू शकता अशा शिक्षकाला. गणित स्वतःच सर्वांसाठी खुले आहे, दाखलन झालेले असो किंवा नसो.

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कसे कराल: Parametric equations and polar coordinates

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

लोक विचारतात ते प्रश्न

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.

अधिक माहिती Precalculus

Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theorem