Problem 30 - Contemporary Precalculus

Question

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$2\left(\cos \frac{7 \pi}{6}+i \sin \frac{7 \pi}{6}\right)$$

Step-by-Step Solution

Answer

Question: Convert the complex number in its polar form, \(2\left(\cos \frac{7 \pi}{6}+i \sin \frac{7 \pi}{6}\right)\), into the Cartesian form (a + bi). Answer: -√3 - i

Step 1: Evaluate the trigonometric functions

In order to evaluate the real and imaginary parts of our complex number, we need to evaluate the cosine and sine functions for the given angle (\(\frac{7\pi}{6}\)). Recall that: $$\cos(\frac{7\pi}{6}) = -\frac{\sqrt{3}}{2}$$ $$\sin(\frac{7\pi}{6}) = -\frac{1}{2}$$

Step 2: Multiply the trigonometric values by the magnitude

Now with both cosine and sine values calculated, we must multiply them by the magnitude of 2: $$2\cos(\frac{7\pi}{6})\ = 2\left(-\frac{\sqrt{3}}{2}\right)\ = -\sqrt{3} $$ $$2\sin(\frac{7\pi}{6})\ = 2\left(-\frac{1}{2}\right)\ = -1$$

Step 3: Combine the values into a + bi form

With the real and imaginary parts calculated, we can now express the given complex number in the form a + bi: $$2\left(\cos \frac{7 \pi}{6}+i \sin \frac{7 \pi}{6}\right) = -\sqrt{3} - i$$

Key Concepts

Polar Form

When dealing with complex numbers, polar form is an alternative to the traditional Cartesian form \((a+bi)\). Instead of representing these numbers with a real part \(a\) and an imaginary part \(b\), we can describe a complex number in terms of its

• Magnitude: Often denoted as \(r\). It represents the "length" from the origin to the point in the complex plane. It is computed using \( r = \sqrt{a^2 + b^2} \).
• Angle: Also known as the argument, denoted by \(\theta\), and represents the direction of the number from the positive real axis. It is given by \(\theta = \arctan\left(\frac{b}{a}\right)\).

In polar form, a complex number is expressed as \(r(\cos\theta + i\sin\theta)\). Here, \(r\) is the magnitude and \(\theta\) is the angle. Notice this is the format in which our original problem is given. Polar form is particularly useful for easily multiplying or dividing complex numbers, as you work with magnitudes and just add or subtract the angles.

Trigonometric Functions

Trigonometric functions are fundamental when working with the polar form of complex numbers. In context, these are:

• Cosine: It helps determine the real part of the complex number. For an angle \(\theta\), it is expressed as \(\cos\theta\).
• Sine: Used for finding the imaginary part added to the real part in the form of \(i\sin\theta\).

In our example, to convert a complex number from polar to Cartesian form, we used the trigonometric values:- \(\cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\)- \(\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}\)Thus, together, these functions provide a bridge between polar coordinates and the standard Cartesian coordinates for complex numbers, allowing us to break down complex calculations into manageable steps.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i^2 = -1\). This definition allows the creation of complex numbers, which are numbers that include a real part and an imaginary part.In any complex number \(a+bi\), the term \(bi\) contains the imaginary unit, where \(b\) represents the coefficient of the imaginary part. In the context of our problem, the imaginary unit helps in the transformation from polar form using \(\sin\theta\) to give us the imaginary component.Here’s why it is crucial:- **Visual Representation:** On the complex plane, the imaginary unit forms the vertical axis.- **Operations:** It enables operations that involve roots of negative numbers and is essential in expressing complex solutions in fields like engineering and physics.By understanding the role of \(i\), we can seamlessly express, interpret, and manipulate complex numbers, such as transforming \(-i\sin\theta\) into its real-world application within math and science.