Chapter 10 Summary and Review

Key Concepts

1. 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.

   

2. Non-uniqueness of Polar Coordinates.

   1. 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.
   2. The point [latex](r, \theta)[/latex] can also be designated by [latex](-r, \theta + \pi)\text{.}[/latex]
   3. The pole has coordinates [latex](0, \theta)\text{,}[/latex] for any value of [latex]\theta\text{.}[/latex]
3. 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]

4. Conversion Equations.

   1. To convert from polar coordinates [latex](r, \theta)[/latex] to Cartesian:
      [latex]x = r\cos \theta\\ y = r \sin \theta[/latex]
   2. 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.

5. 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]
6. 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.
7. Standard graphs in polar coordinates include circles and roses, cardioids and limaçons, lemniscates, and spirals.
8. 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.

9. Imaginary Unit.

   We define the imaginary unit, [latex]i[/latex], by

   [latex]i^2=-1~~~~~~\text{or}~~~~~~i=\sqrt{-1}[/latex]

10. The square root of a negative number is an imaginary number: if [latex]a \gt 0,~ \sqrt{-a}=i\sqrt{a}[/latex]
11. A complex number [latex]z[/latex] is the sum of a real number and an imaginary number, [latex]z=a+bi\text{.}[/latex]
12. 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]

13. 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]
14. We can graph complex numbers in the complex plane.
15. We can visualize the sum of two complex numbers by vector addition in the complex plane.

16. 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.

17. The nonreal zeros of a polynomial with real coefficients always occur in conjugate pairs.
18. Multiplying a complex number by [latex]i[/latex] rotates its graph by [latex]90°[/latex] around the origin.

19. 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.

20. 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]

21. 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]

22. 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]