Problem 33 - Precalculus

Question

Use a calculator to express each complex number in polar form. $$-\frac{1}{2}+\frac{3}{4} i$$

Step-by-Step Solution

Answer

The polar form is \\(\frac{\sqrt{13}}{4}(\cos(\theta) + i\sin(\theta))\\). Use a calculator to find \\(\theta\\).

Step 1: Identify the real and imaginary parts

To express the complex number in polar form, first identify the real and imaginary parts. Here, the complex number is \(-\frac{1}{2}+\frac{3}{4} i\). The real part is \(-\frac{1}{2}\) and the imaginary part is \(+\frac{3}{4}\).

Step 2: Calculate the modulus

The modulus of a complex number \(a+bi\) is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\). Thus, calculate \(|z| = \sqrt{\left( -\frac{1}{2} \right)^2 + \left( \frac{3}{4} \right)^2}\). Compute this as \(|z| = \sqrt{\frac{1}{4} + \frac{9}{16}} = \sqrt{\frac{4}{16} + \frac{9}{16}} = \sqrt{\frac{13}{16}}\), which simplifies to \(|z| = \frac{\sqrt{13}}{4}\).

Step 3: Calculate the argument

The argument \(\theta\) of a complex number is calculated using the formula \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). In this case, it's \(\theta = \tan^{-1}\left( \frac{\frac{3}{4}}{-\frac{1}{2}} \right) = \tan^{-1}\left( -\frac{3}{2} \right)\). Use a calculator to determine \(\theta\), ensuring to adjust it to the correct quadrant since the real part is negative and the imaginary part is positive, placing it in the second quadrant.

Step 4: Express the complex number in polar form

Now that we have the modulus \(|z|\) and the argument \(\theta\), the complex number in polar form is given by \(z = |z|(\cos\theta + i\sin\theta)\). Substitute the values into this formula to get the final polar form of the complex number.

Key Concepts

Complex Number Modulus

Before diving into polar representation, it’s essential to understand the modulus of a complex number. The modulus, also known as the absolute value, measures the distance of a complex number from the origin in the complex plane. Given a complex number of the form \( a + bi \) where \( a \) is the real part and \( b \) is the imaginary part, the modulus is found using the formula:

• \(|z| = \sqrt{a^2 + b^2}\)

In other words, the modulus is the hypotenuse of a right triangle where the real and imaginary parts are the sides. For our problem, the real part is \(-\frac{1}{2}\) and the imaginary part is \(\frac{3}{4}\). By plugging these into our formula, we find the modulus:

• \(|z| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{3}{4}\right)^2} = \frac{\sqrt{13}}{4}\)

The modulus not only tells us the distance, but it is also a crucial step toward constructing the polar form.

Complex Number Argument

The argument of a complex number, denoted as \(\theta\), defines the angle formed by the complex number and the positive real axis in the complex plane. To find this angle, we use the inverse tangent function:

• \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\)

With \(a = -\frac{1}{2}\) and \(b = \frac{3}{4}\) from our exercise, we have:

• \(\theta = \tan^{-1}\left(-\frac{3}{2}\right)\)

The computation of \(\theta\) is straightforward with a calculator. However, you must consider the signs. The negative real part and positive imaginary part indicate that our complex number lies in the second quadrant, requiring an adjustment of the argument accordingly. Hence, it's essential to calculate precisely to determine the accurate angle for our polar form.

Polar Coordinates

Polar coordinates are a way of expressing points in the plane using the distance from the origin and an angle from the positive axis, rather than traditional Cartesian coordinates. The polar form of a complex number \( z \) is represented by:

• \( z = |z|(\cos\theta + i\sin\theta) \)

This converts our modulus and argument into a more usable form. In our example, \(|z| = \frac{\sqrt{13}}{4}\) and \(\theta\) is the angle found earlier.

Using these, the complex number \(-\frac{1}{2} + \frac{3}{4}i\) can be expressed in polar form. This transformation is crucial as it allows for a more straightforward interpretation and manipulation of complex numbers, especially in fields like engineering and physics.

Trigonometry

Trigonometry is foundational in understanding the transformation between Cartesian and polar forms. The core trigonometric functions – sine and cosine – are tied to the circle’s geometry, directly applying to the links between complex numbers and their polar forms.

• The cosine function relates to the real part (\(a\)) of the complex number.
• The sine function connects to the imaginary part (\(b\)).

These functions allow us to pinpoint an angle's position, reallocating a complex number from a rectangular grid to a circular format. In this exercise, once the modulus and argument are known, trigonometric identities make it seamless to express the number in terms of \( r(\cos\theta + i\sin\theta) \).

Understanding these relationships enhances your grasp of geometric interpretations of complex numbers, facilitating advanced calculations in applications like electronic circuit theory and signal processing.