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Analytic Trigonometry

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Analytic Trigonometry

Barnett Ziegler Bylean

Polar coordinates and complex numbers

Polar coordinates

- Given (3, 30⁰)
- From unit circle we know that cos(ө)= x/r
- sin(ө) = y/r
- Thus x = 3cos(30⁰) y = 3 sin(30⁰)

- (-3, ) (2, 53⁰)

- Given (,-1) convert to polar coordinate
- r2 = 2 + (-1)2 = 3 + 1 = 4
- r = ± 2
- tan(ө) = ө =
- Thus (2,) or (-2, )

- Uses the same replacements
- Ex : change to polar form
3x2 + 5y = 4 – 3y2

3r2cos2(ө) + 5r sin(ө) = 4 – 3 r2sin2(ө)

3r3 = 4 – 5r sin(ө)

- Ex: change to rectangular form
- r( 3cos(ө) + 7sin(ө)) = 5

Complex numbers

- Z = x + iy then z = rcos(x) + irsin(y)
- In pre - calculus or calculus you will explore the relation between this form of z and the form z = reiө