Finding the Modulus of a Complex Number Expression

The question asks us to find the modulus of the complex number given by the expression \(\rm \frac {\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta}\), where \(\rm i = \sqrt {-1}\).

Understanding Complex Numbers in Polar Form

A complex number \(z = x + iy\) can also be represented in polar form as \(z = r(\cos \phi + i \sin \phi)\), where \(r = |z| = \sqrt{x^2 + y^2}\) is the modulus and \(\phi\) is the argument (angle). Euler's formula states that \(e^{i\phi} = \cos \phi + i \sin \phi\).

Applying Euler's Formula

Using Euler's formula, the numerator and denominator of the given complex number can be written as:

• Numerator: \(\cos \theta + i \sin \theta = e^{i\theta}\)
• Denominator: \(\cos \theta - i \sin \theta\). Note that \(\cos(-\theta) = \cos \theta\) and \(\sin(-\theta) = -\sin \theta\). So, \(\cos \theta - i \sin \theta = \cos(-\theta) + i \sin(-\theta) = e^{-i\theta}\). Alternatively, the denominator is the conjugate of the numerator, and the conjugate of \(e^{i\theta}\) is \(e^{-i\theta}\).

So, the complex number can be written as:

$$z = \frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta} = \frac{e^{i\theta}}{e^{-i\theta}}$$

Simplifying the Complex Number

Using the rules of exponents (\(\frac{a^m}{a^n} = a^{m-n}\)), we can simplify the expression:

$$z = e^{i\theta - (-i\theta)} = e^{i\theta + i\theta} = e^{2i\theta}$$

Converting this back to the rectangular form using Euler's formula \(e^{i\phi} = \cos \phi + i \sin \phi\) with \(\phi = 2\theta\):

$$z = \cos(2\theta) + i \sin(2\theta)$$

Calculating the Modulus

The modulus of a complex number \(z = a + ib\) is given by \(|z| = \sqrt{a^2 + b^2}\). In this case, \(a = \cos(2\theta)\) and \(b = \sin(2\theta)\). So, the modulus is:

$$|z| = |\cos(2\theta) + i \sin(2\theta)| = \sqrt{(\cos(2\theta))^2 + (\sin(2\theta))^2}$$

Using the fundamental trigonometric identity \(\cos^2 x + \sin^2 x = 1\), with \(x = 2\theta\):

$$|z| = \sqrt{\cos^2(2\theta) + \sin^2(2\theta)} = \sqrt{1} = 1$$

Alternative Method using Modulus Properties

Alternatively, we can use the property of moduli that for two complex numbers \(z_1\) and \(z_2\), \(|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}\), provided \(z_2 \neq 0\). In this case, \(z_1 = \cos \theta + i \sin \theta\) and \(z_2 = \cos \theta - i \sin \theta\).

Calculate the modulus of the numerator \(z_1\):

$$|z_1| = |\cos \theta + i \sin \theta| = \sqrt{\cos^2 \theta + \sin^2 \theta} = \sqrt{1} = 1$$

Calculate the modulus of the denominator \(z_2\):

$$|z_2| = |\cos \theta - i \sin \theta| = \sqrt{\cos^2 \theta + (-\sin \theta)^2} = \sqrt{\cos^2 \theta + \sin^2 \theta} = \sqrt{1} = 1$$

Now, calculate the modulus of the fraction:

$$|z| = \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} = \frac{1}{1} = 1$$

Both methods yield the same result.

The modulus of the given complex number \(\rm \frac {\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta}\) is 1.

Revision Table: Modulus of Complex Numbers

Additional Information on Complex Numbers and Modulus

• Complex Number: A number of the form \(a + ib\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with \(i^2 = -1\). \(a\) is the real part and \(b\) is the imaginary part.
• Modulus: The modulus of a complex number represents its magnitude or distance from the origin in the complex plane. It is always a non-negative real number.
• Argument: The argument of a complex number is the angle between the positive real axis and the line segment connecting the origin to the complex number in the complex plane.
• Polar Form: \(z = r(\cos \phi + i \sin \phi)\), where \(r = |z|\) and \(\phi = \arg(z)\). This form is particularly useful for multiplication, division, and powers of complex numbers.
• Euler's Formula: \(e^{i\phi} = \cos \phi + i \sin \phi\). This formula connects complex exponentials to trigonometric functions and is fundamental in understanding the polar form.
• The modulus of \(\cos \theta + i \sin \theta\) is always 1, as it represents a complex number lying on the unit circle in the complex plane. This is because \(\cos^2 \theta + \sin^2 \theta = 1\).