Finding the Modulus of a Complex Number involving Trigonometry

The question asks us to find the modulus of the complex number \(z = \frac{1+\text{i}\sin \theta}{1−\text{i}\sin \theta}\), where i represents the imaginary unit \(\sqrt{-1}\).

Understanding Modulus of a Complex Number

The modulus of a complex number \(a + bi\) is given by \(\sqrt{a^2 + b^2}\). It represents the distance of the complex number from the origin in the complex plane.

For a complex number that is a ratio of two complex numbers, say \(z = \frac{z_1}{z_2}\), the modulus of z is the ratio of the moduli of the numerator and the denominator, i.e., \(|z| = \frac{|z_1|}{|z_2|}\).

Step-by-Step Calculation of the Modulus

Let the numerator be \(z_1 = 1 + \text{i}\sin \theta\) and the denominator be \(z_2 = 1 - \text{i}\sin \theta\).

Calculate the Modulus of the Numerator (\(z_1\))

\(z_1 = 1 + \text{i}\sin \theta\). Here, the real part is \(a=1\) and the imaginary part is \(b=\sin \theta\).

The modulus of \(z_1\) is:

\(|z_1| = \sqrt{1^2 + (\sin \theta)^2}\)

\(|z_1| = \sqrt{1 + \sin^2 \theta}\)

Calculate the Modulus of the Denominator (\(z_2\))

\(z_2 = 1 - \text{i}\sin \theta\). Here, the real part is \(a=1\) and the imaginary part is \(b=-\sin \theta\).

The modulus of \(z_2\) is:

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

\(|z_2| = \sqrt{1 + \sin^2 \theta}\)

Calculate the Modulus of z (\(|z|\))

Now, we find the modulus of z using the formula \(|z| = \frac{|z_1|}{|z_2|}\):

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

Since the numerator and the denominator are equal, the ratio is 1, provided \(\sqrt{1 + \sin^2 \theta} \neq 0\). The term \(\sin^2 \theta\) is always non-negative, so \(1 + \sin^2 \theta\) is always greater than or equal to 1. Thus, \(\sqrt{1 + \sin^2 \theta}\) is never zero.

Therefore, \(|z| = 1\).

Conclusion

The modulus of the given complex number \(z = \frac{1+\text{i}\sin \theta}{1−\text{i}\sin \theta}\) is 1.

Revision Table: Complex Number Modulus Concepts

Additional Information: Properties of Modulus

The modulus of a complex number has several useful properties:

• The modulus of a product is the product of the moduli: \(|z_1 z_2| = |z_1| |z_2|\).
• The modulus of a quotient is the quotient of the moduli: \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\) (for \(z_2 \neq 0\)).
• The modulus of a complex number is equal to the modulus of its conjugate: \(|z| = |\bar{z}|\). For \(z = a+bi\), \(\bar{z} = a-bi\), \(|z| = \sqrt{a^2+b^2}\) and \(|\bar{z}| = \sqrt{a^2+(-b)^2} = \sqrt{a^2+b^2}\).
• The modulus of a power of a complex number: \(|z^n| = |z|^n\).
• \(z \bar{z} = |z|^2\).

In this specific problem, we could also notice that the numerator \(1 + \text{i}\sin \theta\) and the denominator \(1 - \text{i}\sin \theta\) are complex conjugates of each other. Let \(z_1 = 1 + \text{i}\sin \theta\). Then \(z_2 = \bar{z_1}\). The given complex number is \(z = \frac{z_1}{\bar{z_1}}\).

We know that \(|z_1| = |\bar{z_1}|\). Therefore, \(|z| = \left|\frac{z_1}{\bar{z_1}}\right| = \frac{|z_1|}{|\bar{z_1}|} = \frac{|z_1|}{|z_1|} = 1\).