Problem 78 - 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. $$ (1+i)(1-i \sqrt{3})(-\sqrt{3}+i) $$

Step-by-Step Solution

Answer

In the polar form, the answer is \(4\sqrt{2}(cos(\(\pi/12\)) + isin(\(\pi/12\)))\). In the rectangular form, the answer is 6.8 + 1.95i.

Step 1: Conversion to Polar Form

Assume \(a+bi\) as rectangular form, where \(a\) is the real part and \(b\) is the imaginary part. The polar form of complex number is \(r(cos\(\theta\) + i sin\(\theta\))\), where \(r = \sqrt{a^2 + b^2}\) and \(\theta = atan(\(b/a\))\). Using these formulas, the polar form of \(1+i\) will be \(\sqrt{2}(cos(\(\pi/4\)) + isin(\(\pi/4\)))\), of \(1-i\sqrt{3}\) will be \(2(cos(-\(\pi/3\)) + isin(-\(\pi/3\)))\) and of \(-\sqrt{3}+i\) will be \(2(cos(\(\pi/6\)) + isin(\(\pi/6\)))\).

Step 2: Multiplication in Polar Form

For multiplication in polar form, we multiply the magnitudes and add the angles. So, \(\sqrt{2}(cos(\(\pi/4\)) + isin(\(\pi/4\))) * 2(cos(-\(\pi/3\)) + isin(-\(\pi/3\))) * 2(cos(\(\pi/6\)) + isin(\(\pi/6\)))\) = \(\sqrt{2}*2*2 * (cos(\(\pi/4 - \pi/3 + \pi/6\)) + isin(\(\pi/4 - \pi/3 + \pi/6\))) = 4\sqrt{2}(cos(\(\pi/12\)) + isin(\(\pi/12\)))\).

Step 3: Conversion Back to Rectangular Form

The rectangular form of \(4\sqrt{2}(cos(\(\pi/12\)) + isin(\(\pi/12\)))\) is \(4\sqrt{2} cos(\(\pi/12\)) + 4\sqrt{2}isin(\(\pi/12\))i\). Using the values of \(cos(\(\pi/12\))\) and \(sin(\(\pi/12\))\), we get 6.8 + 1.95i.

Key Concepts

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. They are written in the form of \(a + bi\), where:

• \(a\) is the real part.
• \(bi\) is the imaginary part, with \(i\) representing the imaginary unit \(\sqrt{-1}\).

In the expression \((1 + i)(1 - i \sqrt{3})(-\sqrt{3} + i)\), each term is a complex number.
These numbers can be manipulated using regular algebraic rules, but they can also be transformed into polar form to simplify multiplication and division.
Understanding both forms allows us to perform various operations more easily, particularly multiplication, which becomes a matter of multiplying magnitudes and adding angles when in polar form.

Rectangular Form

Rectangular form is the standard way to write complex numbers. It's the \(a + bi\) format that looks like coordinates on a complex plane.

• \(a\) indicates the distance from the origin along the real axis.
• \(bi\) indicates the distance along the imaginary axis.

For the equation given, each term like \(1 + i\) or \(-\sqrt{3} + i\) starts out in rectangular form.
This form allows us to see the real and imaginary components clearly,
which is useful for addition and subtraction. However, operations like multiplication can be cumbersome,which is why converting to polar form can simplify calculations.

Angle Addition

Angle addition is crucial in converting complex numbers from one form to another using trigonometric identities.
When multiplying complex numbers in polar form, we multiply their magnitudes and sum their angles.
The angles, also known as arguments of the complex numbers, can be visualized as rotations on the complex plane. For example, if you have angles \(\theta_1\), \(\theta_2\), and \(\theta_3\) from our numbers:

• The resulting angle is \(\theta_1 + \theta_2 + \theta_3\).
• Using trigonometry, this makes multiplication straightforward, as you just add these angles together.

The final step involves using the aggregate angle to convert back to rectangular form, giving us the solution in its original, relatable state. Understanding angle addition simplifies how we manage complex number multiplications in polar form,
making calculations easier and reducing chances for errors.