Problem 21 - Precalculus with Limits

Question

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(-5i\)

Step-by-Step Solution

Answer

The complex number -5i is represented on the complex plane as a point at coordinates (0, -5). The trigonometric form of the complex number -5i is given by \(5(\cos(-π⁄2) + i \sin(-π⁄2))\).

Step 1: Plotting the complex number

The complex number -5i can be represented on the complex plane with a point at coordinate (0, -5). So a point should be marked at that position.

Step 2: Find the magnitude of the complex number

The magnitude \(r\) of the complex number \(z = a + bi\) is given by \[r = \sqrt{a^2 + b^2}\]. Since our number is -5i or 0-5i, a = 0 and b = -5, hence, \(r = \sqrt{0 + (-5)^2} = 5\].

Step 3: Find the argument of the complex number

The argument \(θ\) is given by \[θ = \arctan(\frac{b}{a})\] if \(a > 0\) and \[θ = \arctan(\frac{b}{a}) + π\] if \(a < 0\). However, if \(a = 0\) and \(b ≠0\), then \(θ = π⁄2\) for \(b > 0\) and \(θ = -π⁄2\) for \(b < 0\). Here, \(b = -5\), so \(θ = -π⁄2\).

Step 4: Write the complex number in trigonometric form

The trigonometric form is given by \(r(\cos(θ) + i \sin(θ))\). Substituting \(r = 5\) and \(θ = -π/2\), the trigonometric form of the complex number is given by \(5(\cos(-π⁄2) + i \sin(-π⁄2))\).

Key Concepts

Complex Plane Representation

In mathematics, a complex number extends our familiar real numbers to include an imaginary part. Each complex number can be expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. To visualize complex numbers, we use the complex plane. This is just like the Cartesian plane, but instead of \(x\) and \(y\) axes, we have the real axis and the imaginary axis.

Representing a complex number on the complex plane involves two steps:

• You plot the real part on the horizontal (real) axis.
• You plot the imaginary part on the vertical (imaginary) axis.

For example, the complex number \(-5i\) means that there is no real part; it's simply "0 – 5i." Therefore, you would plot the point at \((0, -5)\) on the complex plane. This graphical representation helps in understanding how complex numbers add or subtract geometrically.

Magnitude of a Complex Number

Finding the magnitude of a complex number is essential for various calculations, especially when expressing the number in trigonometric form. This magnitude is often referred to as the number's modulus or absolute value. It essentially measures the distance of the complex number from the origin (0, 0) on the complex plane.

To find the magnitude \(r\) of a complex number \(z = a + bi\), use the formula:

• \[ r = \sqrt{a^2 + b^2} \]

Let's take \(-5i\) as our example. Here, \(a = 0\) and \(b = -5\). Plug these into the formula:

• \[ r = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5 \]

The magnitude of \(-5i\) is 5. This tells us how far \(-5i\) is from the origin in the complex plane, almost like measuring how far a shadow might stretch from a light source.

Argument of a Complex Number

Another important aspect of complex numbers is finding their argument or angle, which tells us the direction from the origin to the point representing the number on the complex plane. The argument \(θ\) is the angle in standard position (measured counterclockwise from the positive real axis) to the point \((a, b)\) on the plane.

In trigonometry, the argument is found using the formula:

• \[θ = \arctan\left(\frac{b}{a}\right)\]

But there are exceptions:

• If \(a = 0\), you use \(π/2\) for \(b > 0\) and \(-π/2\) for \(b < 0\).

For \(-5i\), \(a = 0\) and \(b = -5\), so the argument \(θ\) is \(-π/2\).

This information is crucial when converting complex numbers into their trigonometric form because it determines the angle used in the expressions \(\cos(θ)\) and \(\sin(θ)\). By knowing both the magnitude and the argument, we can fully describe the complex number's position and orientation in the plane.