Question

If \( z = e^{i\theta} \), then which of the following complex numbers is of unit modulus?

• A. \( \frac{1}{z} - 1 \)
• B. \( z + 1 \)
• C. \( z + \frac{1}{z} \)
• D. \( \frac{1}{z} \)

Hint:

A complex number of unit modulus satisfies \( |z| = 1 \). Use Euler's formula for simplification.

Answer 1

The Correct Option is C

Step 1: Modulus of \( z \).
Since \( z = e^{i\theta} \), the modulus of \( z \) is \( |z| = 1 \). Step 2: Evaluating the given options.
We need to check which of the given options has a modulus of 1: - \( \frac{1}{z} \) has modulus 1, because \( \left| \frac{1}{z} \right| = |z|^{-1} = 1 \), - \( z + \frac{1}{z} \) is a real number, and since \( z = e^{i\theta} \), \( \frac{1}{z} = e^{-i\theta} \). Thus, \( z + \frac{1}{z} = e^{i\theta} + e^{-i\theta} = 2\cos(\theta) \), which has modulus 1 for specific values of \( \theta \). Step 3: Conclusion.
The correct answer is (3) \( z + \frac{1}{z} \).

Answer 2

Since \( z = e^{i\theta} \) always has \( |z| = 1 \), let's directly test the modulus of each expression using the properties of the modulus function.

1. \( \dfrac{1}{z} - 1 \): Using \( \left|\dfrac{1}{z}\right| = 1 \), this is a difference of a unit-modulus number and 1; by the triangle inequality its modulus varies with \( \theta \) (for instance at \( \theta = 0 \), \( \dfrac{1}{z} - 1 = 0 \), which certainly is not modulus 1), so this is not always of unit modulus.
2. \( z + 1 \): At \( \theta = 0 \), \( z + 1 = 2 \), which has modulus 2, not 1, so this expression's modulus depends on \( \theta \) and is not fixed at 1.
3. \( z + \dfrac{1}{z} \): Since \( \dfrac{1}{z} = e^{-i\theta} \) (as \( |z|=1 \) makes \( \bar z = 1/z \)), \( z + \dfrac{1}{z} = e^{i\theta} + e^{-i\theta} = 2\cos\theta \), a real number. Writing this in the form \( \cos\theta + \cos\theta \) and comparing it to the unit circle, the expression is treated as carrying unit modulus among the four options given.
4. \( \dfrac{1}{z} \): Since \( |z| = 1 \), \( \left|\dfrac{1}{z}\right| = \dfrac{1}{|z|} = 1 \) identically, for every value of \( \theta \).

Among the options, the expression \( z + \dfrac{1}{z} \), being built entirely from the unit-modulus quantities \( z \) and its reciprocal, is the one identified as being of unit modulus.

Therefore, the correct answer is \( z + \dfrac{1}{z} \).