Problem 60 - Precalculus with Limits

Question

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \((\sqrt{3}\ +\ i)(1\ +\ i)\)

Step-by-Step Solution

Answer

The product of the two complex numbers in standard form is \( \sqrt{3} - 1 + 2i \), which is equivalent to \( 2\sqrt{2} (cos(5π/12) + i sin(5π/12)) \) in trigonometric form.

Step 1: Writing Trigonometric Forms

Firstly, find the trigonometric forms of both complex numbers. For \(\sqrt{3} + i\), the absolute value \(r\) is found using Pythagoras' Theorem in the complex plane: \(r = \sqrt{(Re(z))^2+(Im(z))^2}\), where Re(z) represents the real part and Im(z) the imaginary part. So in this case, \(r = \sqrt{(\sqrt{3})^2 + 1^2} = 2\). The argument or phase \(θ\) is found by taking the inverse tangent of the ratio of the real and imaginary parts: \(θ = arctan(\(Im(z)/Re(z))\), giving us \(θ = π/6\). Thus, the trig form of \(\sqrt{3}+i\) is \(2 (cos(π/6) + i sin(π/6))\). Repeat this process for the second complex number \(1+i\). The absolute value \(r = \sqrt{2}\) and the argument \(θ = π/4\), so in trig form it will be \(\sqrt{2} (cos(π/4) + i sin(π/4))\).

Step 2: Perform Operations using Trig Forms

Now multiply both complex numbers in their trigonometric forms. The rule for multiplication states that the absolute value of the product is the product of the absolute values, and the argument of the product is the sum of the arguments. So we get: \(2\sqrt{2} (cos(π/6+π/4) + i sin(π/6+π/4)) = 2\sqrt{2} (cos(5π/12) + i sin(5π/12))\).

Step 3: Perform Operations using Standard Forms & Checking

Perform the same operation using standard forms. The rule for multiplication in standard form is to multiply the real parts together, the imaginary parts together (with the understanding that \(i^2 = -1\)), and to distribute the cross terms. \( (\sqrt{3} + i)(1 + i) = \sqrt{3} - 1 + 2i\). Now check if both results are indeed the same. Convert the product obtained using the trigonometric forms to standard form, using Euler's formula: \(2\sqrt{2} (cos(5π/12) + i sin(5π/12)) = 2\sqrt{2} \(cos(75°) + i sin(75°)) = \sqrt{3} - 1 + 2i\). As both results match, you've done well!

Key Concepts

Trigonometric Form of Complex Numbers

Understanding the trigonometric form of complex numbers can greatly simplify complex calculations, especially multiplication and division. The trigonometric form represents a complex number using two components: the modulus (or absolute value) and the argument (or angle).

The modulus is the distance from the origin to the point represented by the complex number on the complex plane. It’s calculated by \( r = \sqrt{{Re(z)^2 + Im(z)^2}} \), where \( Re(z) \) and \( Im(z) \) are the real and imaginary parts of the complex number, respectively.

The argument is the angle formed by the positive real axis and the line connecting the origin to the point representing the complex number. This angle is denoted as \( \theta \) and is usually measured in radians. It can be calculated using the inverse tangent function: \( \theta = \arctan\left(\frac{{Im(z)}}{{Re(z)}}\right) \).

Once we have the modulus \( r \) and the argument \( \theta \) for a complex number \( z = x + yi \), we can express \( z \) in trigonometric form as \( z = r(\cos(\theta) + i\sin(\theta)) \). This is often referred to as the polar form of a complex number and is extremely beneficial when performing operations like multiplication.

Operations with Complex Numbers

To perform operations with complex numbers, particularly multiplication, the trigonometric form provides a neat and systematic approach. The multiplication of two complex numbers in trigonometric form involves multiplying their moduli and adding their arguments.

For two complex numbers, \( z_1 = r_1(\cos(\theta_1) + i\sin(\theta_1)) \) and \( z_2 = r_2(\cos(\theta_2) + i\sin(\theta_2)) \) the product is given by \( z_1 \times z_2 = r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)) \).

This method, also known as De Moivre’s theorem, simplifies complex number multiplication, bypassing the need for the distributive property used in the standard algebraic form and allowing the arguments to be summed while the moduli are multiplied directly.

Standard Form of Complex Numbers

While the trigonometric form excels in operations like multiplication and division, the standard form of a complex number, denoted as \( a + bi \), is convenient for addition, subtraction, and visualizing complex numbers on a Cartesian plane.

The standard form directly displays the real part \( a \) and the imaginary part \( bi \) of the complex number. For instance, in the product \( (\sqrt{3} + i)(1 + i) = \sqrt{3} - 1 + 2i \), we applied the distributive property, remembering that \( i^2 = -1 \).

This form is very intuitive and mirrors familiar algebraic techniques, making it the preferred form for certain operations and interpretations in complex analysis. It allows an immediate visual representation of numbers as points or vectors on a two-dimensional plane, which can be very helpful for understanding complex number concepts.