Problem 25 - Algebra and Trigonometry

Question

In Problems 25-36, write each complex number in rectangular form. $$ 2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

-1 + i\sqrt{3}

Step 1: Identify the polar form components

The problem is given in polar form: the magnitude is 2 and the angle is \(\frac{2\pi}{3}\).

Step 2: Recall the formula for rectangular form

The rectangular form of a complex number is given by \(r(\cos(\theta) + i\sin(\theta))\). Here, \ r=2 \ and \ \theta= \frac{2\pi}{3}\.

Step 3: Calculate \ \cos(\theta) \

Find \ \cos \left(\frac{2\pi}{3}\right)\. \[ \cos \left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]

Step 4: Calculate \ \sin(\theta) \

Find \ \sin \left(\frac{2\pi}{3}\right)\. \[ \sin \left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

Step 5: Substitute the values into rectangular form

Use the values from Steps 3 and 4 to rewrite the complex number: \[ 2\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right)\]

Step 6: Simplify the expression

Distribute the 2: \[2 \left(-\frac{1}{2}\right) + 2 \left(i \frac{\sqrt{3}}{2}\right) = -1 + i\sqrt{3}\]

Key Concepts

Rectangular Form

Complex numbers can often be represented in two main forms: rectangular form and polar form. The rectangular form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The \(a\) represents the real part, and \(b\) represents the imaginary part. This format is similar to how we might plot points on a Cartesian plane, with the real part corresponding to the x-coordinate and the imaginary part corresponding to the y-coordinate.
In our problem, after converting from polar to rectangular form, we find that the given problem simplifies to: \(-1 + i\sqrt{3}\). The number \(-1\) is the real part, and \(\sqrt{3}\) (multiplied by \(i\)) is the imaginary part.

Polar Form

The polar form of a complex number is another way to represent a complex number, which emphasizes the magnitude and direction rather than the real and imaginary components. It is written as \(r(\cos(\theta) + i\sin(\theta))\), where \(r\) is the magnitude (or modulus) of the number, and \(\theta\) is the angle (or argument) that the number forms with the positive real axis.
In our exercise, the complex number is initially given in polar form: \(2(\cos \frac{2 \pi}{3} + i \sin \frac{2 \pi}{3})\). Here, \(r = 2\) and \(\theta = \frac{2 \pi}{3}\). Converting this to rectangular form involves substituting the values of \(\cos(\theta)\) and \(\sin(\theta)\) and simplifying the expression, which leads us to \(-1 + i \sqrt{3}\).

Trigonometric Functions

Trigonometric functions play a vital role in converting complex numbers from polar form to rectangular form. The cosine function, \(\cos(\theta)\), gives the x-coordinate on the unit circle, and the sine function, \(\sin(\theta)\), gives the y-coordinate.
For our problem, we use the angle \(\theta = \frac{2 \pi}{3}\). The values for the trigonometric functions at this angle are:

• \(\cos \left(\frac{2 \pi}{3}\right) = -\frac{1}{2}\)
• \(\sin \left(\frac{2 \pi}{3}\right) = \frac{\sqrt{3}}{2}\)

These values are then used in the formula for the rectangular form \(r(\cos(\theta) + i \sin(\theta))\) to ultimately get \(-1 + i \sqrt{3}\).