The number of values of `theta in (0, pi]`, such that `(cos theta + i sin theta) (cos 3 theta + i sin 3 theta) (cos 5 theta + i sin 5 theta) (cos 7 theta + i sin 7 theta) (cos 9 theta + i sin 9 theta) = -1` is

Options

A. 11
B. 13
C. 14
D. 16

Answer

To solve the problem, we need to find the number of values of \( \theta \) in the interval \( (0, \pi] \) that satisfy the equation:

\[ (\cos \theta + i \sin \theta)(\cos 3\theta + i \sin 3\theta)(\cos 5\theta + i \sin 5\theta)(\cos 7\theta + i \sin 7\theta)(\cos 9\theta + i \sin 9\theta) = -1 \]

### Step 1: Rewrite the equation in exponential form
Using Euler's formula, we can rewrite each term as follows:

\[ \cos \theta + i \sin \theta = e^{i\theta}, \quad \cos 3\theta + i \sin 3\theta = e^{i3\theta}, \quad \cos 5\theta + i \sin 5\theta = e^{i5\theta}, \quad \cos 7\theta + i \sin 7\theta = e^{i7\theta}, \quad \cos 9\theta + i \sin 9\theta = e^{i9\theta} \]

Thus, the left-hand side becomes:

\[ e^{i\theta} \cdot e^{i3\theta} \cdot e^{i5\theta} \cdot e^{i7\theta} \cdot e^{i9\theta} = e^{i(\theta + 3\theta + 5\theta + 7\theta + 9\theta)} \]

### Step 2: Simplify the exponent
Now, we need to simplify the exponent:

\[ \theta + 3\theta + 5\theta + 7\theta + 9\theta = 25\theta \]

So we have:

\[ e^{i25\theta} = -1 \]

### Step 3: Express \(-1\) in exponential form
The number \(-1\) can be expressed in exponential form as:

\[ -1 = e^{i\pi + 2n\pi} \quad \text{for } n \in \mathbb{Z} \]

### Step 4: Set the exponents equal
Setting the exponents equal gives us:

\[ 25\theta = \pi + 2n\pi \]

### Step 5: Solve for \(\theta\)
Dividing both sides by 25, we find:

\[ \theta = \frac{\pi(1 + 2n)}{25} \]

### Step 6: Determine the range of \(n\)
We need to find values of \(n\) such that \( \theta \) lies in the interval \( (0, \pi] \):

\[ 0 < \frac{\pi(1 + 2n)}{25} \leq \pi \]

### Step 7: Solve the inequalities
1. From \(0 < \frac{\pi(1 + 2n)}{25}\):

\[ 1 + 2n > 0 \implies n \geq 0 \]

2. From \(\frac{\pi(1 + 2n)}{25} \leq \pi\):

\[ 1 + 2n \leq 25 \implies 2n \leq 24 \implies n \leq 12 \]

### Step 8: Determine integer values of \(n\)
Thus, \(n\) can take integer values from 0 to 12, inclusive. This gives us:

\[ n = 0, 1, 2, \ldots, 12 \]

### Step 9: Count the values
The total number of integer values for \(n\) is:

\[ 12 - 0 + 1 = 13 \]

### Conclusion
The number of values of \( \theta \) in the interval \( (0, \pi] \) that satisfy the given equation is:

\[ \boxed{13} \]

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