Express of the following complex numbers in modulus amplidutes form. `1-i`

Answer

To express the complex number \( z = 1 - i \) in modulus-amplitude (polar) form, we follow these steps:

### Step 1: Identify the complex number
The given complex number is:
\[ z = 1 - i \]

### Step 2: Calculate the modulus \( r \)
The modulus \( r \) of a complex number \( z = a + bi \) is given by:
\[ r = \sqrt{a^2 + b^2} \]
Here, \( a = 1 \) and \( b = -1 \). Thus, we calculate:
\[ r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the argument \( \theta \)
The argument \( \theta \) can be found using the formulas:
\[ \cos \theta = \frac{a}{r}, \quad \sin \theta = \frac{b}{r} \]
Substituting the values we have:
\[ \cos \theta = \frac{1}{\sqrt{2}}, \quad \sin \theta = \frac{-1}{\sqrt{2}} \]

### Step 4: Determine the angle \( \theta \)
The cosine is positive and sine is negative, which places \( \theta \) in the fourth quadrant. The reference angle corresponding to \( \cos \theta = \frac{1}{\sqrt{2}} \) and \( \sin \theta = -\frac{1}{\sqrt{2}} \) is \( 45^\circ \). Therefore, we find:
\[ \theta = 360^\circ - 45^\circ = 315^\circ \]

### Step 5: Write the polar form
The polar form of the complex number is expressed as:
\[ z = r (\cos \theta + i \sin \theta) \]
Substituting the values of \( r \) and \( \theta \):
\[ z = \sqrt{2} \left( \cos 315^\circ + i \sin 315^\circ \right) \]

### Final Answer
Thus, the modulus-amplitude form of the complex number \( 1 - i \) is:
\[ z = \sqrt{2} \left( \cos 315^\circ + i \sin 315^\circ \right) \]

---