Precalculus by Richard Wright

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The way of the sluggard is blocked with thorns, but the path of the upright is a highway. Proverbs‬ ‭15‬:‭19‬ ‭NIV‬‬‬‬

6-06 Trigonometric Form of a Complex Number

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.5.6

circuit board
Figure 1: Circuit board with capacitors. credit (wikimedia/Christian Taube)

Complex numbers are mathematically interesting, but can be used to solve real-world problems such as in electrical engineering to find how different electrical components effect electrical current.

Graph Complex Numbers

Complex numbers were introduced in lesson 2-01 as solutions to polynomial equations. Recall that the complex unit is \(i = \sqrt{-1}\) and that complex numbers are written in the form a + bi.

To graph complex numbers, a rectangular coordinate system is used with the horizontal axis for the real part and the vertical axis for the imaginary part. Numbers are graphed by finding the point with the real and imaginary parts. For example, the points 2 + 3i and −1 + 2i.

Figure 2: Graph of 2 + 3i and −1 + 2i.
Graph a Complex Number

The complex plane has the real axis as the horizontal and the imaginary axis as vertical.

Plot a number by finding the point with the specified real and imaginary parts.

Example 1: Graph Complex Numbers

Graph the complex numbers 1 – 2i, −3 + i, −2, and 3i.

Solution

Move the distance of the real number to the right and then up the imaginary part.

Figure 3: Graph of 1 – 2i, −3 + i, −2, and 3i.
Try It 1

Graph 2 + 3i and −1 − 2i.

Answer

Absolute value is defined as the distance a number is from 0. For complex numbers the distance formula needs to be used.

$$\lvert a + bi \rvert = \sqrt{a^2 + b^2}$$

Notice that i is not used to find the absolute value because the distance formula uses the horizontal and vertical distances, in this case a and b.

Figure 4: Absolute value of a + bi.
Absolute Value of a Complex Number

$$\lvert a + bi \rvert = \sqrt{a^2 + b^2}$$

Example 2: Absolute Value

Find the absolute value of (a) 1 – 2i and (b) 3 + i.

Solution
  1. $$\lvert a + bi \rvert = \sqrt{a^2 + b^2}$$

    $$\lvert 1 - 2i \rvert = \sqrt{1^2 + \left(-1\right)^2}$$

    $$= \sqrt{5}$$

  2. $$\lvert a + bi \rvert = \sqrt{a^2 + b^2}$$

    $$\lvert 3 + i \rvert = \sqrt{3^2 + 1^2}$$

    $$= \sqrt{10}$$

Try It 2

Find the absolute value of 4 − 3i.

Answer

5

Trigonometric Form of a Complex Number

Another way to graph a complex number is by the distance from the origin and the angle in standard position.

Figure 5: Graph of r(cos θ + i sin θ).

From the graph, a = r cos θ and b = r sin θ.

$$z = a + bi$$

$$z = r \cos θ + ir \sin θ$$

$$z = r\left(\cos θ + i \sin θ\right)$$

This is called the trigonometric form or polar form.

Also from the graph \(r = \sqrt{a^2 + b^2}\) and \(\tan θ = \frac{b}{a}\).

Trigonometric Form of a Complex Number

$$z = r\left(\cos θ + i\sin θ\right)$$

r is called the modulus and θ is called the argument

Convert between trigonometric form and standard form using

$$a = r \cos θ$$

$$b = r \sin θ$$

$$r = \sqrt{a^2 + b^2}$$

$$\tan θ = \frac{b}{a}$$

Example 3: Write a Complex Number in Trigonometric Form

Write (a) −2 + 5i and (b) 12 – 5i in trigonometric form.

Solution
  1. Figure 6: -2 + 5i

    Start by finding r.

    $$r = \sqrt{a^2 + b^2}$$

    $$= \sqrt{\left(-2\right)^2 + 5^2}$$

    $$= \sqrt{29}$$

    Now find θ.

    $$\tan θ = \frac{b}{a}$$

    $$\tan θ = \frac{5}{-2}$$

    $$θ ≈ -1.1903 + π$$

    $$θ ≈ 1.95$$

    The π was added to put the angle in the correct quadrant. Now write the number.

    $$z = \sqrt{29}\left(\cos 1.95 + i \sin 1.95\right)$$

  2. Figure 6: 12 − 5i

    Start by finding r.

    $$r = \sqrt{a^2 + b^2}$$

    $$= \sqrt{12^2 + \left(-5\right)^2}$$

    $$= 13$$

    Now find θ.

    $$\tan θ = \frac{b}{a}$$

    $$\tan θ = \frac{-5}{12}$$

    $$θ = -0.3948 + 2π$$

    $$θ = 5.8884$$

    The 2π was added to make the angle positive. Now write the number.

    $$z = 13\left(\cos 5.89 + i \sin 5.89\right)$$

Try It 3

Write 4 − 4i in trigonometric form.

Answer

\(4\sqrt{2}\left(\cos \frac{7π}{4} + i \sin \frac{7π}{4}\right)\)

Example 4: Write a Complex Number in Standard Form

Write (a) \(4\left(\cos \frac{2π}{3} + i \sin \frac{2π}{3}\right)\) and (b) \(10\left(\cos \frac{π}{4} + i \sin \frac{π}{4}\right)\) in standard form.

Solution
  1. Evaluate the trigonometric expressions.

    $$4\left(\cos \frac{2π}{3} + i \sin \frac{2π}{3}\right)$$

    $$4\left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)$$

    Now distribute the 4.

    $$-2 + 2\sqrt{3} i$$

  2. Evaluate the trigonometric expressions.

    $$10\left(\cos \frac{π}{4} + i \sin \frac{π}{4}\right)$$

    $$10\left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{3}}{2}\right)$$

    Now distribute the 10.

    $$5\sqrt{2} + 5\sqrt{2} i$$

Try It 4

Write \(16\left(\cos \frac{11π}{6} + i \sin \frac{11π}{6}\right)\) in standard form.

Answer

\(8\sqrt{3} - 8i\)

Lesson Summary

Graph a Complex Number

The complex plane has the real axis as the horizontal and the imaginary axis as vertical.

Plot a number by finding the point with the specified real and imaginary parts.


Absolute Value of a Complex Number

$$\lvert a + bi \rvert = \sqrt{a^2 + b^2}$$


Trigonometric Form of a Complex Number

$$z = r\left(\cos θ + i\sin θ\right)$$

r is called the modulus and θ is called the argument

Convert between trigonometric form and standard form using

$$a = r \cos θ$$

$$b = r \sin θ$$

$$r = \sqrt{a^2 + b^2}$$

$$\tan θ = \frac{b}{a}$$

Helpful videos about this lesson.

Practice Exercises (*Optional)

  1. In your own words, explain why i is not part of the absolute value of a complex number formula.
  2. How do you calculate the absolute value of a complex number in trigonometric form?
  3. Graph the following complex numbers.

  4. −3 – 4i
  5. 2 + 5i
  6. 4
  7. \(2\left(\cos \frac{π}{2} + i \sin \frac{π}{2}\right)\)
  8. Find the absolute value of the complex numbers.

  9. −3 – 4i
  10. 2 + 5i
  11. 3(cos 35˚ + i sin 35˚)
  12. Write the following complex numbers in standard form.

  13. 88(cos π + i sin π)
  14. \(5\left(\cos \frac{5π}{4} + i \sin \frac{5π}{4}\right)\)
  15. \(12\left(\cos \frac{11π}{6} + i \sin \frac{11π}{6}\right)\)
  16. Write the following complex numbers in trigonometric form.

  17. 3i
  18. \(-7\sqrt{2} + 7\sqrt{2} i\)
  19. 24 – 7i
  20. Mixed Review

  21. (6-05) Are the vectors parallel, orthogonal, or neither: \(\langle -1, 2 \rangle\) and \(\langle 4, 2 \rangle\)?
  22. (6-05) Find the angle between the vectors \(\langle -1, 2 \rangle\) and \(\langle -2, -4 \rangle\).
  23. (6-04) A hiker in the woods hikes 1.5 miles at N 20˚ W, then turns and hikes 5 miles due east. Where is the hiker from his starting point?
  24. (6-03) Write \(\langle 6, 2 \rangle\) in linear combination form.
  25. (6-02) Find the area of ΔBCD where b = 25, c = 7, and d = 24.

Answers

  1. Absolute value is the distance from the origin, so you have to use the distance formula. The horizontal and vertical distances for the distance formula are real, not imaginary.
  2. |z| = r (the modulus)
  3. 5
  4. \(\sqrt{29}\)
  5. 3
  6. −88
  7. \(-\frac{5\sqrt{2}}{2} - \frac{5\sqrt{2}}{2}i\)
  8. \(6\sqrt{3} - 6i\)
  9. \(3\left(\cos \frac{π}{2} + i \sin \frac{π}{2}\right)\)
  10. \(14\left(\cos \frac{3π}{4} + i \sin \frac{3π}{4}\right)\)
  11. 25(cos 6.00 + i sin 6.00)
  12. orthogonal
  13. 126.87°
  14. 4.70 mi at E 17.44° N
  15. \(6\hat{i} + 2\hat{j}\)
  16. 84