Precalc Polar Coordinates (Unit 14)

r * cos (θ) + i * r * sin (θ) or r * cis(θ) (short form for equation above)

A complex number can be represented in polar form as _______________

sum or difference between a real number and an imaginary number z = a + bi ; a and b are real numbers and i is imaginary and equals sqareroot of (-1)

Complex numbers are the __________ and have the form ________

z(with a bar over top of z) = a - bi sort of like a reflection or mirror image of a complex number

Conjugate of a complex number has the form _________

360 degrees (angle measure with opposite rotation) coterminal angle for 45 degrees is -215

Coterminal angles are found by adding or subtracting ____________

multiplying both complex numbers by the conjugate of the bottom (a+bi)/(c+di) = (a+bi)/(c+di) * (c-di)/(c-di) = (ac + bd)(bc-ad)i/(c^2+d^2)

Divide complex numbers by ___________

the same point described in different ways (add or subtract 360 degrees or 2pi. Can also use negative r values with 180 degree or pi)

Equivalent points are _____________

Distance from the pole (origin)

In polar coordinates r is ______________

degrees of counterclockwise rotation from 0 degrees (due east)

In polar coordinates θ is ____________

A real number (a + bi) (a - bi) = a^2 + b^2

Multiplying a complex number by conjugate gives ____________

(r, θ)

Polar coordinates are given in ______________

vertical line test You can describe shapes and figures as functions that would not be possible in rectangular coordinates

Polar equations are immune to the ________________

law of exponents

Powers of complex numbers in polar form also obey the ____________

1. face the x axis 2 rotate θ degrees counterclockwise 3 walk r units along given angle ( if r is negative take backward steps or turn 180 degrees and walk in the opposite direction along same angle + 180)

Process for plotting polar coordinates 1. face the ___________ 2. rotate ___________ degrees _____________ 3. walk __________ units along the given angle

cartesian xy plane x axis for real numbers y axis for imaginary numbers

The complex plane uses the ______________ to graph with _______________ used for real numbers and the ___________ used for imaginary numbers

Rose curve with n petals

The graph of r = a cos (nθ) and r = a sin (nθ) for n = odd integer produce a

Rose curve with 2n petals

The graph of r = cos (nθ) and r = a sin (nθ) for n = even integer produce a ________

Archimedes Spiral

The graph of r = θ (for theta greater than or equal to 0) produce a ________

Lemniscate

The graph of r^2 = a^2 cos (2θ) and r^2 = a^2 sin (2θ) with a not equal to 0 produce a ________

a circle in quadrants I and IV

The graph of r = a cos θ produces __________

a circle in quadrants I and II

The graph of r = a sin θ produces __________

limacon in a cardiod/heart shape

The graphs of r = a +/- b cos θ, r = a +/- b sin θ with a = b for a>0 and b>0 all produce a ____________

limacon with an inner loop

The graphs of r = a +/- b cos θ, r = a +/- b sin θ, with a < b for a>0 and b>0 all produce a ____________

limacon with no inner loop

The graphs of r = a +/- b cos θ, r = a +/- b sin θ, with a > b for a>0 and b>0 all produce a ____________

limacon

The graphs of r = a +/_ b cos θ, r = a +/_ b sin θ all produce a ____________

add real components together and add imaginary components together (a + bi) +(c + di) = (a+c) + (c+d)i (a+bi) - (c + di) = (a-c) + (c-d) i

To add or subtract complex numbers ________

x = r cos (θ) y = r sin (θ) to create (x, y)

To convert from polar coordinate (r, θ) to rectangular coordinates __________

r = sq. root (x^2 + y^2) θ = arctan (y/x) to create (r, θ)

To convert from rectangular coordinate (x, y) to polar coordinate ___________

r e^iθ (uses Euler's formula e^iθ = cosθ + i*sinθ with θ in radians

True polar form for imaginary number is _______

same quadrant Quadrant I and II may have same angle in calculations but be off by 180 degrees Quadgrant II and IV may have same angle in calculations but be off by 180 degrees

When converting from rectangular to polar coordinates make sure to check that the point fall in the ______________________

FOIL - first, outside, inside, last (a + bi)(c+ di) = ac + (ad)i + (bc)i - bd

When multiplying complex numbers use the _______________ method

even power value is 1 or -1 odd power value is i or -i

if i is raised to an even power the value is _____ if i is raised to an odd power the value is ____