If `z= 1+ cos ""(pi)/( 5) + isin""(pi)/(5)`, then what is `|z|` is equal to ?

Options

A. `2 cos ""(pi)/(5)`
B. `2 sin ""(pi)/(5)`
C. `2 cos ""(pi)/(10)`
D. `2 sin ""(pi)/(10)`

Answer

To find the modulus of the complex number \( z = 1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right) \), we will follow these steps:

### Step 1: Identify the real and imaginary parts of \( z \)
The complex number \( z \) can be expressed as:
\[ z = x + iy \]
where \( x = 1 + \cos\left(\frac{\pi}{5}\right) \) and \( y = \sin\left(\frac{\pi}{5}\right) \).

### Step 2: Write the formula for the modulus of \( z \)
The modulus of a complex number \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

### Step 3: Substitute the values of \( x \) and \( y \)
Substituting \( x \) and \( y \) into the modulus formula, we get:
\[ |z| = \sqrt{\left(1 + \cos\left(\frac{\pi}{5}\right)\right)^2 + \left(\sin\left(\frac{\pi}{5}\right)\right)^2} \]

### Step 4: Expand the expression
Now, we will expand the expression inside the square root:
\[ |z| = \sqrt{\left(1 + \cos\left(\frac{\pi}{5}\right)\right)^2 + \sin^2\left(\frac{\pi}{5}\right)} \]
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ = \sqrt{1^2 + 2 \cdot 1 \cdot \cos\left(\frac{\pi}{5}\right) + \cos^2\left(\frac{\pi}{5}\right) + \sin^2\left(\frac{\pi}{5}\right)} \]

### Step 5: Apply the Pythagorean identity
Using the identity \( \cos^2\theta + \sin^2\theta = 1 \):
\[ = \sqrt{1 + 2\cos\left(\frac{\pi}{5}\right) + 1} \]
\[ = \sqrt{2 + 2\cos\left(\frac{\pi}{5}\right)} \]

### Step 6: Factor out the common term
Factoring out the 2:
\[ = \sqrt{2(1 + \cos\left(\frac{\pi}{5}\right))} \]

### Step 7: Use the cosine double angle identity
Using the identity \( 1 + \cos\theta = 2\cos^2\left(\frac{\theta}{2}\right) \):
\[ = \sqrt{2 \cdot 2\cos^2\left(\frac{\pi}{10}\right)} = \sqrt{4\cos^2\left(\frac{\pi}{10}\right)} \]

### Step 8: Simplify the expression
Taking the square root:
\[ = 2\cos\left(\frac{\pi}{10}\right) \]

### Final Answer
Thus, the modulus \( |z| \) is:
\[ |z| = 2\cos\left(\frac{\pi}{10}\right) \]

---