Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and leave the result in trigonometric form. $$\frac{5(\cos 2 \pi+i \sin 2 \pi)}{4(\cos \pi+i \sin \pi)}$$

Step-by-Step Solution

Answer

The result of the operation in trigonometric form is \(-5/4 = 5/4(\cos(\pi) + i\sin(\pi))\).

Step 1: Identify the Polar Form

The given expression is of the form \(r (\cos \phi + i \sin \phi)\). This is the polar form of complex numbers where r represents modulus and \(\phi\) is the argument.

Step 2: Simplify the given complex numbers

Simplify the complex numbers using the identity \(\cos 2\pi + i\sin 2\pi = 1\) and \(\cos \pi + i\sin \pi = -1\). This gives us \(5/4 * 1/-1 = -5/4\).

Step 3: Leave the result in trigonometric form

Write -5/4 in trigonometric form. A complex number a can be written as \(|a|(\cos(\arg a) + i\sin(\arg a))\). In this case, the complex number is real so the trigonometric form is \(-5/4 = 5/4(\cos(\pi) + i\sin(\pi))\).

Key Concepts

Polar Form

Polar form is a way to express complex numbers. Instead of using the standard rectangular form, which involves real and imaginary parts like most people are used to, polar form highlights the modulus (distance from the origin) and the argument (angle with the positive real axis). A complex number in polar form looks like this: - \( r(\cos \phi + i\sin \phi) \). - Here, \( r \) is the modulus and \( \phi \) is the argument.Why use polar form? It's especially handy for multiplication and division of complex numbers. Multiplying them in polar form means multiplying their moduli and adding their arguments. Similarly, dividing involves dividing the moduli and subtracting the arguments. This can simplify operations significantly.For example, if you have two complex numbers in polar form, the product would be: \[g_1 g_2(\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)) \]Understanding the polar form can be a powerful tool when tackling complex number operations.

Modulus and Argument

The modulus and argument are key elements in understanding polar form.

• The modulus, denoted as \(r\), is the distance from the origin to the point representing the complex number in the complex plane. It is calculated using the formula \( r = \sqrt{a^2 + b^2} \), where \(a\) and \(b\) are the real and imaginary parts, respectively.
• The argument, denoted as \(\phi\), is the angle the line connecting the origin to the point makes with the positive real axis. You find it using the formula \( \phi = \tan^{-1}(\frac{b}{a}) \).

For our exercise: - The modulus of the numerator (\(5(\cos 2 \pi + i \sin 2 \pi)\)) and denominator (\(4(\cos \pi + i \sin \pi)\)) are 5 and 4, respectively. - Therefore, the modulus of the result becomes \( \frac{5}{4} \).The arguments here, \(2\pi\) and \(\pi\), make the real parts revert to 1 and -1, guiding the simplification process in our polar form calculations.

Trigonometric Form

Trigonometric form, often referred to as polar form, is another way to express complex numbers using trigonometric functions.The standard trigonometric form is: \[ r(\cos \phi + i \sin \phi) \]After finding the modulus and argument, you can express any complex number this way. It particularly helps when you need to convert back into rectilinear form (\(a + ib\)) or when the need arises to simplify operations.In this exercise, after simplifying the complex numbers using known identities, we reach a trigonometric form:- The expression \(-5/4 = 5/4(\cos(\pi) + i\sin(\pi))\).This final step ensures the complex number is neatly presented in a form that easily depicts its real and imaginary components through familiar trigonometric functions, aiding both visualization and comprehension.