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Chapter 6: The Polar System

Difficulty Level: At Grade Created by: CK-12
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Chapter Outline

Chapter Summary

Chapter Summary

In this chapter we made the connection between complex numbers and trigonometry. First, we started with the polar system, by graphing and converting equations into polar coordinates. This allowed us to compare the complex plane with the polar plane and we realized that there are many similarities. Because of this, we are able to convert complex numbers into polar, or trigonometric, form. Converting complex numbers to polar form makes it easier to multiply and divide complex numbers by using the Product and Quotient theorems. These theorems lead to De Moivre’s Theorem, which is a shortcut for raising complex numbers to different powers. Finally, we were able manipulate De Moivre’s Theorem to find all the complex solutions to different equations.


In the complex number \begin{align*}r(\cos \theta + i \sin \theta)\end{align*}, the argument is the angle \begin{align*}\theta\end{align*}.
In the complex number \begin{align*}r(\cos \theta + i \sin \theta)\end{align*}, the modulus is \begin{align*}r\end{align*}. It is the distance from the origin to the point \begin{align*}(x, y)\end{align*} in the complex plane.
Polar coordinate system
A method of recording the position of an object by using the distance from a fixed point and an angle consisting of a fixed ray from that point. Also called a polar plane.
In a polar coordinate system, it is the fixed point or origin.
Polar axis
In a polar coordinate system, it is the horizontal ray that begins at the pole and extends in a positive direction.
Polar coordinates
The coordinates of a point plotted on a polar plane \begin{align*}(r, \theta)\end{align*}.
Polar Equation
An equation which uses polar coordinates.
Polar Form
Also called trigonometric form is the complex number \begin{align*}x + yi\end{align*} written as \begin{align*}r (\cos \theta + i \sin \theta)\end{align*} where \begin{align*}r=\sqrt{x^2+y^2}\end{align*} and \begin{align*}\tan \theta=\frac{y}{x}\end{align*}.

Review Questions

  1. Plot \begin{align*}A\left(-3, \frac{3\pi}{4}\right)\end{align*} and find three other equivalent coordinates.
  2. Find the distance between \begin{align*}(2, 94^\circ)\end{align*} and \begin{align*}(7, -73^\circ)\end{align*}.
  3. Graph the following polar curves.
    1. \begin{align*}r=3 \sin 5 \theta\end{align*}
    2. \begin{align*}r=6-3 \cos \theta\end{align*}
    3. \begin{align*}r=2+5 \cos 9 \theta\end{align*}
  4. Determine the equations of the curves below.
  5. Convert each equation or point into polar form.
    1. \begin{align*}A(-6, 11)\end{align*}
    2. \begin{align*}B(15, -8)\end{align*}
    3. \begin{align*}C(9, 40)\end{align*}
    4. \begin{align*}x^2+(y-6)^2=36\end{align*}
  6. Convert each equation or point into rectangular form.
    1. \begin{align*}D\left(4, -\frac{\pi}{3}\right)\end{align*}
    2. \begin{align*}E(-2, 135^\circ)\end{align*}
    3. \begin{align*}r=7\end{align*}
    4. \begin{align*}r=8 \sin \theta\end{align*}
  7. Determine where \begin{align*}r=6+5 \sin \theta\end{align*} and \begin{align*}r=3-4 \cos \theta\end{align*} intersect.
  8. Change \begin{align*}-3 + 8i\end{align*} into polar form.
  9. Change \begin{align*}15 \angle 240^\circ\end{align*} into rectangular form.
  10. Multiply or divide the following complex numbers using polar form.
    1. \begin{align*}\left(7cis \frac{7\pi}{4}\right) \cdot \left(3cis \frac{\pi}{3}\right)\end{align*}
    2. \begin{align*}\frac{8 \angle 80^\circ}{2 \angle -155^\circ}\end{align*}
  11. Expand \begin{align*}[4\left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)]^6\end{align*}
  12. Find the \begin{align*}6^{th}\end{align*} roots of -64 and graph them in the complex plane.
  13. Find all the solutions of \begin{align*}x^4+32=0\end{align*}.

Review Answers

  1. \begin{align*}A \left(-3, \frac{3\pi}{4}\right)\end{align*} three equivalent coordinates \begin{align*}\rightarrow \left(3, -\frac{\pi}{4}\right), \left(3, \frac{7\pi}{4}\right), \left(-3, -\frac{5\pi}{4}\right)\end{align*}.
  2. \begin{align*}(2, 94^\circ)\end{align*} and \begin{align*}(7, -73^\circ)\end{align*} \begin{align*}d &=\sqrt{2^2+7^2-2(2)(7)\cos (94^\circ-(-73^\circ))}\\ &=\sqrt{4+49-28 \cos 167^\circ}\\ &=\sqrt{80.28} \approx 8.96\end{align*}
  3. a) \begin{align*}r=3 \sin 5 \theta\end{align*} b) \begin{align*}r=6-3 \cos \theta\end{align*} c) \begin{align*}r=2+5 \cos 9 \theta\end{align*}
    1. \begin{align*}r = 2 - 6\cos \theta\end{align*}
    2. \begin{align*}r = 7 + 3\sin \theta\end{align*}
  4. a. \begin{align*}A(-6, 11) \rightarrow r=\sqrt{36+121} \approx 12.59, \tan \theta=-\frac{11}{6}, \theta=118.6^\circ \rightarrow (12.59, 118.6^\circ)\end{align*} b. \begin{align*}B(15, -8) \rightarrow r=\sqrt{225+64} = 17, \tan \theta=-\frac{8}{15}, \theta=-28.1^\circ \rightarrow (17, -28.1^\circ)\end{align*} c. \begin{align*}C(9, 40) \rightarrow r=\sqrt{91+1600} = 41, \tan \theta=\frac{40}{9}, \theta=77.3^\circ \rightarrow (41, 77.3^\circ)\end{align*} d. \begin{align*}x^2+(y-6)^2 &= 36\\ r^2 \cos^2 \theta+(r \sin \theta-6)^2 &= 36\\ r^2 \cos^2 \theta+r^2 \sin^2 \theta-12r \sin \theta+36 &= 36\\ r^2-12r \sin \theta &= 0 \ \text{or}\\ r^2 &= 12r \sin \theta\\ r &= 12 \sin \theta\end{align*}
  5. a. \begin{align*}D\left(4, -\frac{\pi}{3}\right) \rightarrow x=4 \cos \left(-\frac{\pi}{3}\right)=2, y=4 \sin \left(-\frac{\pi}{3}\right)=-2\sqrt{3} \rightarrow (2, -2 \sqrt{3})\end{align*} b. \begin{align*}E(-2, 135^\circ) \rightarrow x=-2 \cos 135^\circ = \sqrt{2}, y=-2 \sin 135^\circ=-\sqrt{2} \rightarrow (\sqrt{2}, -\sqrt{2})\end{align*} c. \begin{align*}r=7 \rightarrow r^2=49 \rightarrow x^2+y^2=49\end{align*} d. \begin{align*}r &= 8 \sin \theta\\ r^2 &= 8r \sin \theta\\ x^2+y^2 &= 8y\\ y^2-8y &= -x^2\\ y^2-8y+16 &= 16-x^2\\ (y-4)^2 &= 16-x^2\\ x^2 + (y-4)^2 &= 16\end{align*}
  6. \begin{align*}r=6+5 \sin \theta\end{align*} and \begin{align*}r = 3-4 \cos \theta\end{align*} \begin{align*}^*\end{align*} angle measures in the graph are in radians Note: The two determined points of intersection [\begin{align*}(6.91, 2.95)\end{align*} and \begin{align*}(1.44, -1.16)\end{align*}] were estimated from the trace function on a graphing calculator and are not precise solutions for either equation.
  7. \begin{align*}-3+8i, x=-3, y=8 \rightarrow r=\sqrt{(-3)^2+8^2} \approx 8.54, \tan \theta =-\frac{8}{3} \rightarrow \theta = 110.56^\circ \ 8.54(\cos 110.56^\circ+i \sin 110.56^\circ\end{align*}
  8. \begin{align*}15 \angle 240^\circ, r = 15, \theta = 240^\circ \rightarrow x=15 \cos 240^\circ=-7.5, y=15 \sin 240^\circ=-\frac{15 \sqrt{3}}{2}=-7.5 \sqrt{3}\end{align*} So, \begin{align*}15 \angle 240^\circ = -7.5 - 7.5i \sqrt{3}\end{align*}.
    1. \begin{align*}\left(7cis \frac{7\pi}{4}\right) \cdot \left(3cis \frac{\pi}{3}\right)=21cis \left(\frac{7\pi}{4}+\frac{\pi}{3}\right)=21cis \frac{25\pi}{12}\end{align*}
    2. \begin{align*}\frac{8 \angle 80^\circ}{2 \angle -155^\circ}=4 \angle (80^\circ-(-155^\circ))=4 \angle 235^\circ\end{align*}
  9. \begin{align*}\left[4\left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)\right]^6 = 4^6 \left(\cos \frac{6\pi}{4}+i \sin \frac{6\pi}{4}\right)=4096 \left(\cos \frac{3\pi}{2}+i \sin \frac{3\pi}{2}\right)\end{align*}
  10. -64 in polar form is \begin{align*}64 (\cos \pi - i \sin \pi)\end{align*} \begin{align*}& [64(\cos (\pi + 2\pi k )+i \sin (\pi+2\pi k))]^{\frac{1}{6}}\\ & 2 \left(\cos \left(\frac{\pi+2\pi k}{6}\right)+i \sin \left(\frac{\pi+2\pi k}{6}\right)\right)\\ & 2 \left(\cos \left(\frac{\pi}{6}+\frac{\pi k}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{\pi k}{3}\right)\right)\\ & z_1=2\left(\cos\left(\frac{\pi}{6}+\frac{0\pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{0\pi}{3}\right)\right)=2\cos \frac{\pi}{6}+2i \sin \frac{\pi}{6}=\frac{2\sqrt{3}}{2}+\frac{2i}{2}=\sqrt{3}+i\\ & z_2=2\left(\cos\left(\frac{\pi}{6}+\frac{\pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{\pi}{3}\right)\right)=2\cos \frac{\pi}{2}+2i \sin \frac{\pi}{2}=2i\\ & z_3=2\left(\cos\left(\frac{\pi}{6}+\frac{2\pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{2\pi}{3}\right)\right)=2\cos \frac{5\pi}{6}+2i \sin \frac{5\pi}{6}=-\frac{2\sqrt{3}}{2}+\frac{2i}{2}=-\sqrt{3}+i\\ & z_4=2\left(\cos\left(\frac{\pi}{6}+\pi\right)+i \sin \left(\frac{\pi}{6}+\pi\right)\right)=2\cos \frac{7\pi}{6}+2i \sin \frac{7\pi}{6}=-\frac{2\sqrt{3}}{2}-\frac{2i}{2}=-\sqrt{3}-i\\ & z_5=2\left(\cos\left(\frac{\pi}{6}+\frac{4\pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{4\pi}{3}\right)\right)=2\cos \frac{3\pi}{2}+2i \sin \frac{3\pi}{2}=-2i\\ & z_6=2\left(\cos\left(\frac{\pi}{6}+\frac{5\pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{5\pi}{3}\right)\right)=2\cos \frac{11\pi}{6}+2i \sin \frac{11\pi}{6}=\frac{2\sqrt{3}}{2}-\frac{2i}{2}=\sqrt{3}-i\end{align*} Graph of the solutions:
  11. \begin{align*}& x^4+32=0 \rightarrow x^4=-32+0i=-32(\cos \pi+i \sin \pi)\\ & \qquad [32(\cos (\pi+2\pi k)+i \sin (\pi+2\pi k))]^{\frac{1}{4}}\\ & \qquad 2 \sqrt[4]{2} \left(\cos \left(\frac{\pi+2\pi k}{4}\right)+i \sin \left(\frac{\pi+2\pi k}{4}\right)\right)\\ & \qquad 2 \sqrt[4]{2}\left(\cos \left(\frac{\pi}{4}+\frac{\pi k}{2}\right)+i \sin \left(\frac{\pi}{4}+\frac{\pi k}{2}\right)\right)\\ & z_1=2 \sqrt[4]{2}\left(\cos \left(\frac{\pi}{4}\right)+i \sin \left(\frac{\pi}{4}\right)\right)=2 \sqrt[4]{2}\cos \frac{\pi}{4}+2i \sqrt[4]{2} \sin \frac{\pi}{4}=\frac{2\sqrt[4]{2}\sqrt{2}}{2}+\frac{2i\sqrt[4]{2}\sqrt{2}}{2}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ =\sqrt[4]{2}^3+i \sqrt[4]{2}^3\\ & z_2=2 \sqrt[4]{2}\left(\cos \left(\frac{\pi}{4}+\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{4}+\frac{\pi}{2}\right)\right)=2\sqrt[4]{2}\cos \frac{3\pi}{4}+2i \sqrt[4]{2} \sin \frac{3\pi}{4}=-\frac{2\sqrt[4]{2}\sqrt{2}}{2}+\frac{2i\sqrt[4]{2}\sqrt{2}}{2}\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ =-\sqrt[4]{2}^3+i \sqrt[4]{2}^3\\ & z_3=2 \sqrt[4]{2}\left(\cos \left(\frac{\pi}{4}+\pi \right)+i \sin \left(\frac{\pi}{4}+\pi \right)\right)=2\sqrt[4]{2}\cos \frac{5\pi}{4}+2i \sqrt[4]{2} \sin \frac{5\pi}{4}=-\frac{2\sqrt[4]{2}\sqrt{2}}{2}-\frac{2i\sqrt[4]{2}\sqrt{2}}{2}\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad =-\sqrt[4]{2}^3-i \sqrt[4]{2}^3\\ & z_4=2 \sqrt[4]{2}\left(\cos \left(\frac{\pi}{4}+\frac{3\pi}{2}\right)+i \sin \left(\frac{\pi}{4}+\frac{3\pi}{2}\right)\right)=2\sqrt[4]{2}\cos \frac{7\pi}{4}+2i \sqrt[4]{2} \sin \frac{7\pi}{4}=\frac{2\sqrt[4]{2}\sqrt{2}}{2}-\frac{2i\sqrt[4]{2}\sqrt{2}}{2}\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad =\sqrt[4]{2}^3-i \sqrt[4]{2}^3\end{align*}

Texas Instruments Resources

In the CK-12 Texas Instruments Trigonometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9704.

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