Chapter 10: Polar Coordinates and Complex Numbers
Chapter 10 Summary and Review
Key Concepts
-
Polar Coordinates.
The polar coordinates of a point [latex]P[/latex] in the plane are [latex](r, \theta)\text{,}[/latex] where
- [latex]\lvert r \rvert[/latex] is the distance from to the pole,
- [latex]\theta[/latex] is the angle measured counterclockwise from the polar axis to the ray through [latex]P[/latex] from the pole.
-
Non-uniqueness of Polar Coordinates.
- Any point with polar coordinates [latex](r, \theta)[/latex] also has coordinates [latex](r, \theta + 2k\pi)\text{,}[/latex] where [latex]k[/latex] is an integer.
- The point [latex](r, \theta)[/latex] can also be designated by [latex](-r, \theta + \pi)\text{.}[/latex]
- The pole has coordinates [latex](0, \theta)\text{,}[/latex] for any value of [latex]\theta\text{.}[/latex]
- In the polar plane, the coordinate grid lines are circles centered at the pole, with equations [latex]r=k\text{,}[/latex] and lines through the pole, with equations [latex]\theta = k\text{.}[/latex]
-
Conversion Equations.
- To convert from polar coordinates [latex](r, \theta)[/latex] to Cartesian:
[latex]x = r\cos \theta\\ y = r \sin \theta[/latex]
- To convert from Cartesian coordinates [latex](x,y)[/latex] to polar:
[latex]r = \sqrt{x^2+y^2}\\ \tan \theta = \dfrac{y}{x}[/latex]
where the choice of [latex]\theta[/latex] depends on the quadrant.
- To convert an equation from Cartesian to polar coordinates, we replace each [latex]x[/latex] with [latex]r\cos \theta[/latex] and each with [latex]y[/latex] with [latex]r\sin \theta\text{.}[/latex] To convert an equation from polar to Cartesian coordinates, look for expressions of the form [latex]r\cos \theta,~r\sin \theta,~r^2\text{,}[/latex] or [latex]\tan \theta\text{.}[/latex]
- When graphing an equation in polar coordinates, we think of sweeping around the pole in the counterclockwise direction, and at each angle [latex]\theta[/latex] the [latex]r[/latex]-value tells us how far the graph is from the pole.
- Standard graphs in polar coordinates include circles and roses, cardioids and limaçons, lemniscates, and spirals.
- To find the intersection points of the polar graphs [latex]r=f(\theta)[/latex] and [latex]r=g(\theta)[/latex] we solve the equation [latex]f(\theta)=g(\theta)\text{.}[/latex] In addition, we should always check whether the pole is a point on both graphs.
-
Imaginary Unit.
We define the imaginary unit, [latex]i[/latex], by
[latex]i^2=-1~~~~~~\text{or}~~~~~~i=\sqrt{-1}[/latex]
- The square root of a negative number is an imaginary number: if [latex]a \gt 0,~ \sqrt{-a}=i\sqrt{a}[/latex]
- A complex number [latex]z[/latex] is the sum of a real number and an imaginary number, [latex]z=a+bi\text{.}[/latex]
- We can perform the four arithmetic operations on complex numbers.
Operations on Complex Numbers.
[latex]z_1+z_2=(a+bi)+(c+di)=(a+c)+(b+d)i \\ z_1-z_2=(a+bi)-(c+di)=(a-c)+(b-d)i \\ z_1z_2=(a+bi)(c+di) = (ac-bd)+(ad+bc)i \\ \dfrac{z_1}{z_2} = \dfrac{a+bi}{c+di} = \dfrac{a+bi}{c+di} \cdot \dfrac{c-di}{c-di} = \dfrac{ac+bd}{c^2+d^2} + \dfrac{bc-ad}{c^2+d^2}i[/latex]
- The product of a nonzero complex number and its conjugate is always a positive real number.
[latex]z \bar{z} = (a+bi)(a-bi) = a^2 - b^2i^2 = a^2 - b^2(-1)=a^2+b^2\text{.}[/latex]
- We can graph complex numbers in the complex plane.
- We can visualize the sum of two complex numbers by vector addition in the complex plane.
-
Fundamental Theorem of Algebra.
Let [latex]p(x)[/latex] be a polynomial of degree [latex]n \ge 1\text{.}[/latex] Then [latex]p(x)[/latex] has exactly [latex]n[/latex] complex zeros.
- The nonreal zeros of a polynomial with real coefficients always occur in conjugate pairs.
- Multiplying a complex number by [latex]i[/latex] rotates its graph by [latex]90°[/latex] around the origin.
-
Polar Form for a Complex Number.
The complex number [latex]z=a+bi[/latex] can be written in the polar form
[latex]z=r(\cos\theta+i\sin \theta)[/latex]
where
[latex]r=\sqrt{a^2+b^2}[/latex]
and [latex]\theta[/latex] is defined by
[latex]a=r \cos \theta,~~~~b=r\sin \theta,~~~~0 \le \theta \le 2\pi[/latex]
The angle [latex]\theta[/latex] is called the argument of the complex number, and [latex]r[/latex] is its length, or modulus.
-
Product and Quotient in Polar Form.
If [latex]z_1=r(\cos \alpha+i\sin \alpha)[/latex] and [latex]z_2=R(\cos \beta+i\sin \beta)\text{,}[/latex] then
[latex]z_1z_2=rR(\cos (\alpha + \beta) + i \sin (\alpha + \beta))[/latex]
and
[latex]\dfrac{z_1}{z_2}=\dfrac{r}{R}(\cos (\alpha - \beta) + i \sin (\alpha - \beta))[/latex]
-
De Moivre’s Theorem.
If [latex]z=r(\cos\alpha+i\sin \alpha)[/latex] is a complex number in polar form, and [latex]n[/latex] is a positive integer, then
[latex]z^n= r^n(\cos n\alpha+i\sin n\alpha)[/latex]
-
Roots of a Complex Number.
A complex number [latex]z=r(\cos \alpha + i\sin \alpha)[/latex] in polar form has [latex]n[/latex] complex [latex]n[/latex]th roots, given by
[latex]z_k = r^{1/n}\left(\cos \dfrac{\alpha + 2\pi k}{n} + i\sin \dfrac{\alpha + 2\pi k}{n}\right)[/latex]
for [latex]k = 0,~1,~2, \cdots,~ n-1\text{.}[/latex]