Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Find all indicated roots and express them in rectangular form. Check your results with a calculator. The square roots of \(4\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)\).

Step-by-Step Solution

Answer

The square roots in rectangular form are \(1 + i\sqrt{3}\) and \(-1 - i\sqrt{3}\).

Step 1: Express in Polar Form

The number is given as \(4(\cos 120^\circ + i \sin 120^\circ)\), which is already in polar form where \(r = 4\) and \(\theta = 120^\circ\).

Step 2: Apply the Square Root Formula

For a complex number in polar form \(r(\cos\theta + i\sin\theta)\), its square roots are found using the formula \(\sqrt{r}\left(\cos\frac{\theta + 360^\circ k}{2} + i\sin\frac{\theta + 360^\circ k}{2}\right)\), where \(k = 0, 1\). Since \(r = 4\), \(\sqrt{r} = 2\).

Step 3: Calculate the First Root

Set \(k = 0\) in the formula to find the first root: \(2\left(\cos \frac{120^\circ}{2} + i \sin \frac{120^\circ}{2}\right) = 2\left(\cos 60^\circ + i \sin 60^\circ\right) = 2\left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = 1 + i\sqrt{3}\).

Step 4: Calculate the Second Root

Set \(k = 1\) in the formula to find the second root: \(2\left(\cos \frac{120^\circ + 360^\circ(1)}{2} + i \sin \frac{120^\circ + 360^\circ(1)}{2}\right) = 2\left(\cos 240^\circ + i \sin 240^\circ\right) = 2\left(-\frac{1}{2} - i \frac{\sqrt{3}}{2}\right) = -1 - i\sqrt{3}\).

Step 5: Convert to Rectangular Form

We found two square roots: \(1 + i\sqrt{3}\) and \(-1 - i\sqrt{3}\), both of which are already in rectangular form. Use a calculator to verify that both are valid square roots.

Key Concepts

Polar Form

Polar form is a way of expressing complex numbers in terms of their magnitude and angle. This form is particularly beneficial when multiplying or dividing complex numbers or finding roots. A complex number in polar form is represented as \[ r(\cos \theta + i \sin \theta) \]where:

• \( r \) is the magnitude of the complex number
• \( \theta \) is the angle measured from the positive real axis to the line representing the complex number in the complex plane.

For example, the complex number used in the exercise, \( 4(\cos 120^{\circ} + i \sin 120^{\circ}) \), has a magnitude \( r = 4 \) and an angle \( \theta = 120^{\circ} \). This form allows us to use De Moivre's Theorem to easily find powers and roots of complex numbers.

Rectangular Form

Rectangular form, also known as the Cartesian form, is another way of representing complex numbers. In this form, a complex number is expressed as:\[ a + bi \] where:

• \( a \) is the real part of the complex number
• \( b \) is the imaginary part of the complex number.

The rectangular form directly reflects the position of the complex number on the complex plane, with the real part on the x-axis and the imaginary part on the y-axis.
In the given exercise, the solutions \(1 + i\sqrt{3}\) and \(-1 - i\sqrt{3}\) are both examples of complex numbers expressed in rectangular form. This form is often more intuitive to use in addition and subtraction of complex numbers.

Square Roots of Complex Numbers

Finding square roots of complex numbers involves a combination of algebra and geometry. By expressing the complex number in polar form, we can apply De Moivre’s Theorem to find the roots. For a complex number given as \( r(\cos \theta + i \sin \theta) \), the square roots use the formula:\[ \sqrt{r}\left(\cos\frac{\theta + 360^{\circ} k}{2} + i\sin\frac{\theta + 360^{\circ} k}{2}\right) \] where:

• \( \sqrt{r} \) is the magnitude of the root
• Different values of \( k \) (in this case \( k = 0 \) and \( k = 1 \)) represent the distinct roots.

Within our example, the number \( 4(\cos 120^{\circ} + i \sin 120^{\circ}) \) produced two roots, \( 1 + i\sqrt{3} \) and \( -1 - i\sqrt{3} \), in rectangular form. Each result represents one of the two square roots, showing the symmetry of roots in the complex plane.