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Chapter 7: Problem 18

Write each complex number in trigonometric form, using degree measure for the argument. $$-7$$

Short Answer

Expert verified
The trigonometric form of -7 is \(7 (\cos(180^{\circ}) + i \sin(180^{\circ}))\) .

Step by step solution

01

Identify the complex number

Given the complex number is $$-7$$ . This number is purely real and negative.
02

Determine the modulus

The modulus of a complex number $$a + bi$$ is given by $$|z| = \sqrt{a^2 + b^2}$$ . Here, the complex number is $$-7 + 0i$$ . Therefore, the modulus is: $$|z| = \sqrt{(-7)^2 + 0^2} = 7$$.
03

Find the argument

For a purely real number, the argument can be determined based on its position on the real axis. Since $$-7$$ lies on the negative real axis, its argument is $$180^{\circ}$$ (or $$\pi$$ radians).
04

Write in trigonometric form

A complex number in trigonometric form is written as $$r (\cos(\theta) + i \sin(\theta))$$ , where $$r$$ is the modulus and $$\theta$$ is the argument. Using the determined values for the modulus and argument: $$7 (\cos(180^{\circ}) + i \sin(180^{\circ}))$$ .

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Key Concepts

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

headline of the respective core concept
Understanding complex numbers involves exploring their standard form and their representation in the trigonometric form. The trigonometric form of a complex number provides a different but very insightful perspective on complex numbers, especially useful in multiplication, division, and finding powers and roots.
Firstly, let's grasp what the trigonometric form is.
headline of the respective core concept

Trigonometric Form of a Complex Number

The trigonometric form of a complex number is a way to express the number using its magnitude (also called modulus) and angle (also called argument). It's written as: \[ r (\text{cos} \theta + i\text{sin} \theta) \] Here, \( r \) is the modulus of the complex number and \( \theta \) (theta) is the argument.
=> The modulus shows how far the number is from the origin on the complex plane.
=> The argument indicates the direction of the number from the origin.
When you use trigonometric form, it becomes easier to visualize and compute operations like multiplication and division of complex numbers.
headline of the respective core concept

Modulus of a Complex Number

Finding the modulus is like finding the length of the number from the origin on the complex plane. For a complex number \( z = a + bi \), the modulus is given by: \[ |z| = \sqrt{a^2 + b^2} \] In our example, the complex number was \( -7 \), which is purely real and can be written as \( -7 + 0i \).
So, the modulus is:\[ |z| = \sqrt{(-7)^2 + 0^2} = 7 \]
This means the distance from the origin to the point \( -7 \) on the complex plane is 7 units.
headline of the respective core concept

Argument of a Complex Number

The argument of a complex number is the angle it makes with the positive real axis. It's denoted by \( \theta \). This is especially easy to find for real numbers.
For our number \( -7 \), which lies on the negative real axis, the angle is straight across the real axis, giving an argument of \( 180^{\circ} \) or \( \pi \) radians.
Having found both the modulus (7) and the argument (\( 180^{潞} \) here), we can now express the number in trigonometric form as:\[ 7 \left( \text{cos}180^{\circ} + i \text{sin} 180^{\circ} \right) \]This helps us use complex numbers more conveniently in various mathematical operations.

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Most popular questions from this chapter

Solve each problem. A river is \(2000 \mathrm{ft}\) wide and flowing at 6 mph from north to south. A woman in a canoe starts on the eastern shore and heads west at her normal paddling speed of \(2 \mathrm{mph} .\) In what direction (measured clockwise from north) will she actually be traveling? How far downstream from a point directly across the river will she land?

Find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\) for each pair of complex numbers, using trigonometric form. Write the answer in the form \(a+b i\). $$z_{1}=3 i, z_{2}=1+i$$

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