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University of Miami

Mathematics

MTH 108

Lun- Yi Tsai

Fall

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MTH 108
Precalculus Mathematics II
Chapter 8 Study Guide Notes
L. Sterling
Abstract
Provide a generalization to each of the key terms listed in this section.
Pole
What is a pole?
A pole is a point in the polar coordinate system.
Polar Axis
What is a polar axis?
A polar axis is a ray with a vertex at the pole in the polar coordinate system.
Polar and Rectangular Coordinates
What are the polar coordinates?
Polar Coordinates are ordered pairs in the form of (r;▯).
(r;▯)
Any point with (r;▯) as their respected polar coordinates, which has ▯ in radians, can represent
by either (r;▯ + 2▯k) or even (▯r;▯ + ▯ + 2pik), which notes that k is any integer. On the other
hand, (r;▯) are also the poles polar coordinates, which notes that ▯ being any possible angle.
What are the rectangular coordinates?
Polar Coordinates are ordered pairs in the form of (x;y).
Polar to Rectangular Coordinates
What are the diﬀerence between polar and rectangular coordinates?
P = Point
Rectangular Coordinates = Polar=Cartesian Coordinates
(x; y) = (r; ▯)
x = r cos ▯
y = r sin ▯
1 Converting Rectangular to Polar Coordinates
With Given Point on a Coordinate Axis
▯ What are the steps for converting from rectangular/cartesian to polar coordinates with the
given point on a coordinate axis?
– Plot the point, which will be labeled (x;y).
– Make a note on which quadrant that the point is lying.
– Make a note on which coordinate axis that the point is lying.
– Find the distance, which is r, from the origin to the given point.
– Find ▯.
With Given Point on a Quadrant
▯ What are the steps for converting from rectangular/cartesian to polar coordinates with the
given point on a quadrant?
– Plot the point, which will be labeled (x;y).
– Make a note on which quadrant that the point is lying.
– Make a note on which coordinate axis that the point is lying.
– Find the distance, which is r, from the origin to the given point by using the following:
p
r = x + y2
▯ If x = 0, then ﬁnd r.
▯ If y = 0, then ﬁnp r.
▯ If x 6= 0, then r =x + y .
p
▯ If y 6= 0, then r =x + y .
– Find ▯.
▯ If x = 0, then ﬁnd ▯.
▯ If y = 0, then ﬁnd ▯.
▯ If x 6= 0, then ﬁnd the quadrant the point lies.
▯ If y 6= 0, then ﬁnd the quadrant the point lies.
▯ When either x 6= 0 or y 6= 0, then the following occurs:
▯y▯
Quadrant I : ▯ = tan1
x
▯ y
Quadrant II : ▯ = ▯ + tan1
x
▯ ▯
Quadrant III : ▯ = ▯ + tan1 y
x
▯1 ▯y▯
Quadrant IV : ▯ = tan
x
Rectangular to Polar
p
2 2
r = x + y
r = x + y 2
y
tan ▯ = x [x 6= 0]
2 Theorem: Horizontal and Vertical Lines
Horizontal Lines
a : Real Number
Equation : r sin ▯ = a
a ▯ 0 : a units lying above the pole
a < 0 : jaj units below the pole
Vertical Lines
a : Real Number
Equation : r cos ▯ = a
a ▯ 0 : a units lying to the right of the pole
a < 0 : jaj units to the left of the pole
Theorem: Equations and Circles
r = 2a sin ▯
Radius : a
Center : (0; a)
r = ▯2a sin ▯
Radius : a
Center : (0; ▯a)
r = 2a cos ▯
Radius : a
Center : (a; 0)
r = ▯2a cos ▯
Radius : a
Center : (▯a; 0)
Circles
Every circle would be passing through the pole:
3 Theorem: Symmetry Tests
Symmetric to X-axis: Polar Axis
Replace With
▯ ▯▯
Symmetric to Y -axis: Line
Replace With
▯ ▯ ▯ ▯
Symmetric to Origin: Pole
Replace With
r ▯r
Logarithmic Spiral
A logarithmic spiral is an equation that can be written like ▯ = n ln(r) as an example with its
own spirals while inﬁnite while both away from and towards the pole.
Complex Numbers
Any numbers that are in the form of
z = a + bi
where the following occur:
▯ a and b are real numbers
▯ z is the standard form
– a is the real part
– b is the imaginary part
▯ i is the imaginary unit
– So this notes that
2
i = i ▯ i = ▯1
Compositions of Complex Numbers
Equality of Complex Numbers
▯
a + bi = c + di
– If and only if a = c and b = d
Sum of Complex Numbers
▯
(a + bi) + (c + di) = (a + c) + (b + d)i
Diﬀerence of Complex Numbers
▯
(a ▯ bi) + (c ▯ di) = (a ▯ c) + (b ▯ d)i
4 Product of Complex Numbers
▯
(a + bi) ▯ (c + di) = (ac ▯ bd) + (ad + bc)i
Quotient of Complex Numbers
a + bi
c + di
(a + bi)(c ▯ di)
=
(c + di)(c ▯ di)
(ac ▯ b(▯d)) + (a(▯d) + bc)i
= c + d2
ac + bd ▯ adi + bci
= 2 2
c + d
ac + bd ▯ adi + bci
=
c + d 2
ac + bd ▯ad + bc
= c + d2 + c + d 2 i
Conjugates
Complex Number and Conjugate
▯ The conjugate of any real number is just the real number.
▯ If z = a + bi is the complex number, then its conjugate, which can be denoted withand
deﬁned with the following:
z
= a + bi
= a ▯ bi
▯ The product of a complex number, which is z = a+bi, and its conjugate, which is z = a▯bi
is actually a nonnegative real number, which makes the following:
(a + bi)(a ▯ bi)
= a(a) + a(▯bi) + bi(a) + bi(▯bi)
2 2 2
= a ▯ abi + abi ▯ b i
2 ▯ 2
= a + (1 ▯ 1)abi ▯ b (▯1)
2 2
= a + (01)abi + b
= a + b 2
Complex Number’s Product and Conjugates
▯ The conjugate’s conjugate of a complex number is just the origi

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