Problem 77 - Algebra and Trigonometry

Question

In Exercises \(77-80,\) convert to polar form and then perform the indicated operations. Express answers in polar and rectangular form. $$ i(2+2 i)(-\sqrt{3}+i) $$

Step-by-Step Solution

Answer

The final expression in rectangular form is \(4\sqrt{2}\cos\left(\frac{5\pi}{12}\right) + 4i\sqrt{2}\sin\left(\frac{5\pi}{12}\right)\).

Step 1 - Convert to Polar Form

Let's do the conversion of numbers into polar form. First, convert \(2 + 2i\) and \(-\sqrt{3} + i\) into polar form. For \(2 + 2i\), the modulus is \(\sqrt{2^2 + 2^2} = 2\sqrt{2}\), and the argument (angle \(\theta\)) is \(\arctan\left(\frac{2}{2}\right) = \arctan(1) = \frac{\pi}{4}\). So in polar form, \(2 + 2i = 2\sqrt{2}\left(\cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right)\right)\). Similarly, for \(-\sqrt{3} + i\), the modulus is \(\sqrt{(-\sqrt{3})^2 + 1^2} = 2\), and the argument is \(\arctan\left(\frac{-1}{\sqrt{3}}\right) = -\frac{\pi}{6}\). So in polar form, \(-\sqrt{3} + i = 2\left(\cos\left(-\frac{\pi}{6}\right) + i \sin\left(-\frac{\pi}{6}\right)\right)\).

Step 2 - Multiply the Complex Numbers

Now, multiply the numbers in polar form including \(i\). The magnitude of the result is the product of the magnitudes, and the angle is the sum of the angles. So, \(i(2 + 2i)(-\sqrt{3} + i) = i \times 2\sqrt{2} \times 2 \times \left[\cos\left(\frac{\pi}{4} - \frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{4}- \frac{\pi}{6}\right)\right]\) = 4\sqrt{2}\left[\cos\left(\frac{\pi}{12}\right) + i \sin\left(\frac{\pi}{12}\right)\right]\). The multiplication by \(i\) rotates the argument by \(\frac{\pi}{2}\) (or \(90^\circ\)), hence the answer in polar form is \(4\sqrt{2}\left[\cos\left(\frac{\pi}{12} + \frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{12} + \frac{\pi}{2}\right)\right]\).

Step 3 - Convert to Rectangular Form

Let's convert the result back into rectangular form. So, the final answer is \(4\sqrt{2}\left[\cos\left(\frac{5\pi}{12}\right) + i \sin\left(\frac{5\pi}{12}\right)\right] = 4\sqrt{2}\cos\left(\frac{5\pi}{12}\right) + 4i\sqrt{2}\sin\left(\frac{5\pi}{12}\right)\). The precise decimal values of these expressions can be evaluated using a scientific calculator.

Key Concepts

Polar Form Conversion

Understanding polar form conversion is crucial when dealing with complex numbers. The polar form of a complex number provides a different way to represent it using a modulus (magnitude) and an argument (angle) instead of 'x' and 'y' coordinates on the complex plane.

To convert a complex number from rectangular form, which is the standard form of 'a + bi', to polar form, we use the following steps:

• Calculate the modulus, which is the distance of the point from the origin, using the formula: \( r = \sqrt{a^2 + b^2} \).
• Find the argument, which is the angle formed with the positive real axis, with the formula: \( \theta = \arctan\left(\frac{b}{a}\right) \) for \( a > 0 \). If \( a \) is negative, you'll need to adjust your angle by adding \( \pi \) to the atan result to get the correct quadrant.

To put the number into polar form, simply write it as \( r(\cos(\theta) + i\sin(\theta)) \), where 'r' is the modulus and '\( \theta \)' is the argument.

For our example exercise, \( 2 + 2i \) converts to polar form as \( 2\sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4})) \), and \( -\sqrt{3} + i \) converts to \( 2(\cos(-\frac{\pi}{6}) + i\sin(-\frac{\pi}{6})) \). These forms are essential for performing operations on complex numbers especially multiplication or division, which are more straightforward in polar form than in rectangular form.

Complex Number Multiplication

Complex number multiplication becomes simpler when the numbers are in polar form. This is because the multiplication of two complex numbers in polar form involves multiplying their moduli and adding their arguments, following the identity \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), known as Euler's formula.

Here are the steps for multiplying two complex numbers in polar form:

• Multiply the moduli of the two complex numbers.
• Add the arguments (angles) of the two complex numbers.

The resulting product is a complex number in polar form with a modulus equal to the product of the two moduli and an argument equal to the sum of the two arguments. When the multiplication includes the imaginary unit \(i\), this corresponds to a rotation by \(\frac{\pi}{2}\) radians or 90 degrees, since \(i\) is at an angle of \(\frac{\pi}{2}\) on the complex plane.

In the example, we multiply \(i\) with the previously converted polar forms to get \(4\sqrt{2}(\cos(\frac{\pi}{12}) + i\sin(\frac{\pi}{12}))\), and after considering the rotation by \(i\), it becomes \(4\sqrt{2}(\cos(\frac{5\pi}{12}) + i\sin(\frac{5\pi}{12}))\), which is the product in polar form.

Rectangular Form Conversion

After operating on complex numbers in polar form, we often need to convert them back to rectangular form to interpret or use the result in a standard format. The process for this conversion involves using the cosine and sine functions.

To convert a complex number from polar to rectangular form, we follow these steps:

• Identify the modulus \(r\) and argument \(\theta\) from the polar form.
• Calculate the real part \(x\) as \(r\cos(\theta)\).
• Calculate the imaginary part \(y\) as \(r\sin(\theta)\).

After these calculations, the complex number can be written in rectangular form as \(x + yi\). This form is more intuitive for visualizing complex numbers as points on the complex plane.

In the case of our exercise, to convert \(4\sqrt{2}(\cos(\frac{5\pi}{12}) + i\sin(\frac{5\pi}{12}))\) back to rectangular form, we compute the real part as \(4\sqrt{2}\cos(\frac{5\pi}{12})\) and the imaginary part as \(4\sqrt{2}\sin(\frac{5\pi}{12})i\). The result presents the complex number in a familiar 'a + bi' format, ready for any further operation or analysis needed.