Analytic Trigonometry

Barnett Ziegler Bylean

CHAPTER 7Polar coordinates and complex numbers

CH 7 - SECTION 1Polar coordinates

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(30 ) y = 3 ⁰sin(30 )⁰

Examples: convert to rectangular coordinates (cartesian)

• (-3, ) (2, 53⁰)

Converting rectangular to polar

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

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

CHAPTER 7 – SEC 3Complex numbers

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