Home page for accesible maths Math 101 Chapter 4: Taylor series and complex numbers 4.33 Complex trigonometric series 4.35 Polar form of complex numbers

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4.34 Complex exponentials and the unit circle

Let $x=\cos\theta$ and $y=\sin\theta$ . For real $\theta$ , the point $e^{i\theta}$ lies on the circle with centre zero and radius one in the complex plane, known as the unit circle. As $\theta$ increases, $e^{i\theta}$ moves round the circle with unit speed, anti clockwise.

Example (Some compass points)

e^{i\pi/4}={{1+i}\over{\sqrt{2}}};\qquad e^{i\pi/2}=i;\qquad e^{3\pi i/4}={{-1% +i}\over{\sqrt{2}}};\qquad e^{i\pi}=-1;

e^{5i\pi/4}=\qquad\quad;\quad e^{6i\pi/4}=\qquad\quad;\quad e^{7i\pi/4}=\qquad% \quad;\quad e^{8\pi i/4}=\qquad\qquad.