Analytic Trigonometry
Barnett Ziegler Bylean
Polar coordinates and complex numbers
CHAPTER 7
Polar coordinates
CH 7 - SECTION 1
Converting a point
polar to rectangular
• 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)
Examples: convert to rectangular
coordinates (cartesian)
• (-3,
7𝜋
)
6
(2, 53⁰)
Converting rectangular to polar
• Given ( 3,-1) convert to polar coordinate
• r2 = 32 + (-1)2 = 3 + 1 = 4
• r=±2
•
•
−1
tan(ө) =
3
−𝜋
Thus (2, )
3
−𝜋
ө=
3
or (-2,
2𝜋
3
)
Converting equations
• 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
CHAPTER 7 – SEC 3
Complex plane-Cartesian coordinates
•
Trig form of complex number
• Z = x + iy
then z = rcos(x) + i rsin(y)
• In pre - calculus or calculus you will explore
the relation between this form of z and the
form z = reiө