Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in the trigonometric form \(r(\cos \theta+i \sin \theta),\) where \(r\) is exact and \(0^{\circ} \leq \theta<360^{\circ}\) $$-2 i$$

Step-by-Step Solution

Answer

The trigonometric form is \\(2(\cos 270^\circ + i\sin 270^\circ)\\).

Step 1: Identify the Real and Imaginary Parts

For the complex number \(-2i\), the real part is \(0\) and the imaginary part is \(-2\).

Step 2: Calculate the Modulus

The modulus \(r\) of a complex number \(a + bi\) can be calculated using \(r = \sqrt{a^2 + b^2}\). In our case, \(-2i\) implies \(a = 0\) and \(b = -2\). The modulus is \(|-2| = 2\).

Step 3: Find the Argument

The argument \(\theta\) of a complex number can be found using the arctan function, but since our number is purely imaginary, located on the negative imaginary axis, \(\theta = 270^\circ\).

Step 4: Write in Trigonometric Form

Now that we have the modulus \(r = 2\) and the angle \(\theta=270^\circ\), the trigonometric form of the complex number is \(2(\cos 270^\circ + i\sin 270^\circ)\).

Key Concepts

Real and Imaginary Parts

Understanding the real and imaginary parts of a complex number is foundational in grasping complex numbers. A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is referred to as the real part, and \(b\) is the imaginary part. For example, in the complex number \(-2i\), the real part is \(0\) because there is no typical "real" component visible (like "\(a\)"), and the imaginary part is \(-2\), associated with the imaginary unit \(i\). This distinction allows us to understand where a complex number falls on the complex plane, with the real part represented on the horizontal axis and the imaginary part on the vertical axis. Identifying these components is the first step in transforming a complex number into its other forms, such as the trigonometric form.

Modulus of a Complex Number

The modulus of a complex number is akin to its length or magnitude when depicted on the complex plane. It tells us how far the number is from the origin \((0,0)\) in this plane. For a complex number \(a + bi\), the modulus is determined using the formula: \(r = \sqrt{a^2 + b^2}\). This is reminiscent of the Pythagorean theorem. For example, consider the purely imaginary number \(-2i\), which can be expressed as \(0 + (-2)i\). Here, the modulus is \(r = \sqrt{0^2 + (-2)^2} = 2\). This value of \(2\) indicates the distance from the origin to the point \(0, -2\) on the complex plane. The modulus is crucial for writing complex numbers in trigonometric form and is always a non-negative real number.

Argument of a Complex Number

The argument of a complex number is the angle it makes with the positive real axis, usually measured in degrees or radians. It helps to pinpoint the number's direction from the origin on the complex plane. In general, to find the argument \(\theta\) of a complex number \(a + bi\), trigonometric functions like arctan are used. However, exceptions exist, such as when the number is purely imaginary. For \(-2i\), the direction from the origin is exactly down the negative imaginary axis. Hence, the angle \(\theta\) is \(270^\circ\), corresponding to being directly south on the unit circle. This insight into angles is fundamental to expressing complex numbers in their trigonometric forms, as it marries the concept of direction with magnitude.