Download 7.7 Polar Coordinates Name: 7.8 De Moivre`s

7.7 Polar Coordinates
7.8 De Moivre's Theorem
Name: _______________
Objectives: Students will be able to convert points and equations
from polar coordinates to rectangular and vice versa. Students will be
able to perform computations with complex numbers in polar form and
use De Moivre's Theorem.
When surveyors record the location of objects using distances
and angles, they are using ____________________.
Mar 23­7:45 AM
Examples: Graph each point.
1.) P(3, 60o)
2.) Q(-1.5, 7π/6)
3.) R(-2,-135o)
Mar 23­8:00 AM
1
Examples The polar coordinates of a point are given. Find its
rectangular coordinates.
(-4, 5π/4)
Jan 22­6:26 PM
Let the point P have polar coordinates (r, θ) and rectangular
coordinates (x, y). Then:
x = rcosθ
y = rsinθ
tan = y/x
r = x2 + y2
Example Find the rectangular coordinates of the point with the
given polar coordinates. P(3, 5π/6)
Jan 22­6:30 PM
2
Example Polar coordinates of P are given. Find all of its polar
coordinates. P = (1, -π/4)
Jan 22­6:34 PM
Examples Convert the rectangular equation to polar form.
1.) x + y = 1
2.) x2 + y2 - 4x + 6y = 12
Example Convert r = 4sinθ to rectangular form. Graph.
Jan 22­6:48 PM
3
Examples: Graph each polar equation. Convert to rectangular
form.
1.) r = 3
2.) θ=3π/4
Mar 23­8:07 AM
Example The location of two ships from Mays Landing Lighthouse,
given in polar coordinates, are (3 mi, 170o) and (5 mi, 150o). Find the
distance between the two ships.
Jan 22­6:52 PM
4
Symmetry
To test for symmetry
1.) about the x-axis
2.) about the y-axis
3.) about the origin
replace
by
Jan 23­7:57 AM
Examples Use the polar symmetry tests to determine symmetry.
1.) r = 1 + 2cosθ
2.) r = 7sin3θ
Jan 23­10:26 AM
5
Types of Polar Curves
Curve
Rose
Lemniscate
Limacon
spiral of
Archimedes
Polar
Equation
r = acosnθ
r = asinnθ
r2 = a2cos2θ
r2 = a2sin2θ
r = a ± bcosθ
r = a ± bsinθ
r = aθ
n is a positive
integer
Cardioid
General
Graph
Jan 23­1:17 PM
Examples Graph the following polar curves. Identify the type of
curve.
2.) r = 6 -5cosθ
1.) r = -3cos4θ
Jan 23­2:15 PM
6
4.) r = 3 - 4sinθ
3.) r = 5 - 5sinθ
5.) r = θ/4
6.) r2 = 9cos2θ
Jan 23­2:15 PM
Complex Number and Complex Plane:
Standard (rectangular) form of a complex number:
Examples Plot u = 1 + 3i, v = 2i , w = -3 - 6i and x = -2 + 4i in the complex plane.
Mar 11­9:40 PM
7
The __________ _____ or ______ of a complex number z = a + bi
is z = a + bi = √a2 + b2 .
In the complex plane, a + bi is the distance of a + bi from the origin.
Example Let z = 2 - 4i. Find z .
Jan 27­6:25 PM
The _______________ _______ or (polar form) of the complex number
z = a + bi is _________________, where a = _______, b = _______,
r = __________, and tanθ = _____. The number r is the absolute value or
modulus of z, and θ is the __________ of z. An angle θ for the trig form
of z can always be chosen so that 0 ≤ θ ≤ 2π, although any angle coterminal
with θ could be used. Consequently, the angle θ and the argument of a
complex number z are not unique. Therefore, the trig form of a complex
number is also not unique.
Jan 27­6:28 PM
8
Examples Find the trig (polar) form of the complex number where
the argument satisfies 0 ≤ θ < 2π.
1.) √3 + i
2.)
45o
4
z
Jan 27­6:34 PM
Examples Write the complex number in standard (rectangular)
form a + bi.
1.) 8(cos210o + isin210o)
2.) √7(cos5π/6 + isin5π/6)
Jan 27­6:35 PM
9
Product and Quotient of Complex Numbers
Let z1 = r1(cosθ1 + isinθ1) and z2 = r2(cosθ2 + isinθ2). Then
1.)
2.)
Jan 27­6:40 PM
Examples
Find the product and quotient of z1 and z2.
1.) z1 = √2(cos118o + isin118o) and z2 = 0.5[cos(-19o) + isin(-19o)].
2.)
z1 = 3 + 3i and z2 = 2 - 2√3 i
Mar 11­9:59 PM
10
De Moivre's Theorem
Let z = r(cosθ + isinθ) and let n be a positive integer.
Then zn = [r(cosθ + isinθ)]n = rn(cosnθ + isinnθ).
Examples
1.) Find [3(cos(3π/2) + isin(3π/2))]5.
2.) (3 + 4i)20
Jan 27­6:47 PM
A complex number v = a + bi is an _____________ if vn = z.
If z = 1, then vn = 1 and v is said to be an _____________.
If z = r(cosθ + isinθ), then the n distinct complex numbers
n
)
)
√r cos(θ + 2πk) + isin(θ + 2πk) , where k = 0, 1, 2, ..., n-1, are
n
n
the nth roots of the complex number z.
Example Find the cube roots of 2(cosπ/4 + isinπ/4).
Jan 27­6:54 PM
11
2.) Solve v3 = -1.
Jan 27­7:01 PM
3.) Find the sixth roots of unity.
Jan 27­7:05 PM
12
Assignments:
7.7: Pages 693-694: #7-21 odd (Just plot the points. Use
the graph paper on the following page.) , 23-75 every other
odd, 105-108 all
7.8: Pages 703-704: #7-59 every other odd, 69, 70
Jan 27­7:07 PM
Mar 11­9:57 PM
13
Jan 22­7:02 PM
Jan 22­7:02 PM
14