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Chapter 7: Problem 28

Write each complex number in trigonometric form, using degree measure for the argument. $$-2+i$$

Short Answer

Expert verified
The trigonometric form of \(-2 + i\) is \ \sqrt{5}(\cos 153.43^\text{掳} + i\sin 153.43^\text{掳}) \.

Step by step solution

01

- Identify Real and Imaginary Parts

Identify the real part and the imaginary part of the complex number. For the complex number \(-2+i\), the real part \(a\) is \-2\ and the imaginary part \(b\) is \ 1\.
02

- Calculate the Magnitude

Use the formula to find the magnitude of the complex number: \[ r = \sqrt{a^2 + b^2} \]. Substitute \-2\ for \(a\) and \ 1\ for \(b\): \[ r = \sqrt{(-2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]. The magnitude \(r\) is \ \sqrt{5}\.
03

- Calculate the Argument

Calculate the argument \(\theta\) using the formula \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]. However, since the complex number is in the second quadrant, adjust the angle by adding \ 180^\text{掳}\: \[ \theta = \tan^{-1}\left(\frac{1}{-2}\right) + 180^\text{掳} = -26.57^\text{掳} + 180^\text{掳} = 153.43^\text{掳} \].
04

- Write in Trigonometric Form

Combine the magnitude and argument into the trigonometric form: \[ r(\cos\theta + i\sin\theta) \]. Thus, the trigonometric form of \(-2 + i\) is: \ \sqrt{5}(\cos 153.43^\text{掳} + i\sin 153.43^\text{掳}) \.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complex numbers
Complex numbers are numbers that have both a real and an imaginary part. In general, they are written in the form:
  • a + bi
where a represents the real part, and bi represents the imaginary part. Here, i is the imaginary unit with the property that i2 = -1.

For example, in the complex number
  • -2 + i
-2 is the real part, and i (with a coefficient of 1) is the imaginary part. Complex numbers allow for a richer set of operations than real numbers alone, providing a more complete understanding of various mathematical concepts, especially in fields like engineering and physics.
magnitude
The magnitude (or absolute value) of a complex number represents its distance from the origin in the complex plane. It is calculated using the formula:
  • \(r = \sqrt{a^2 + b^2} \)
Where a is the real part, and b is the imaginary part. For example, for the complex number -2 + i:
  • \( r = \sqrt{(-2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5}\)
The magnitude of
  • \( -2 + i \)
is therefore \( \sqrt{5} \). This value tells us how 'far' the complex number is from the origin, much like a radius in a circle.
argument
The argument of a complex number is the angle it makes with the positive real axis in the complex plane. It is calculated using:
  • \( \theta = \tan^{-1} \left(\frac{b}{a}\right) \)
However, the position of the complex number in the plane affects this calculation. For instance, our complex number
  • -2 + i
is in the second quadrant, where we need to adjust the angle. We add
  • \( 180^\text{掳} \)
to the initial angle:
  • \( \theta = \tan^{-1} \left(\frac{1}{-2}\right) + 180^\text{掳} \)
  • \( \theta = -26.57^\text{掳} + 180^\text{掳} = 153.43^\text{掳} \)
This gives us the correct angle for the second quadrant and tells us the direction of the complex number from the positive real axis.
trigonometric form
Expressing a complex number in trigonometric form provides a smooth way to visualize and handle complex operations, especially multiplication and division. The general trigonometric form is:
  • \( r ( \cos\theta + i\sin\theta ) \)
where r is the magnitude, and \theta is the argument. For our example with -2 + i, we determined:
  • r = \sqrt{5}
  • \theta = 153.43^\text{掳}
Therefore, in trigonometric form:
  • \( -2 + i = \sqrt{5} (\cos 153.43^\text{掳} + i\sin 153.43^\text{掳}) \)
This format is particularly useful for multiplying and dividing complex numbers, as it easily combines their magnitudes and sums their arguments, providing clean and precise results.

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