If `sin(6/5x) = 0` and `cos (x/5) = 0` , then

Options

A. `x = (n-5)pi`
B. `x=6(n-1)pi`
C. `x=5(n-(1)/(2))pi`
D. `x=5(n+(1)/(2))pi`

Explanation

To solve the equations \( \sin\left(\frac{6}{5}x\right) = 0 \) and \( \cos\left(\frac{x}{5}\right) = 0 \), we will analyze each equation separately.

### Step 1: Solve \( \sin\left(\frac{6}{5}x\right) = 0 \)

The sine function is equal to zero at integer multiples of \( \pi \):
\[ \frac{6}{5}x = n\pi \quad \text{for } n \in \mathbb{Z} \]
To find \( x \), we rearrange the equation:
\[ x = \frac{5n\pi}{6} \]

### Step 2: Solve \( \cos\left(\frac{x}{5}\right) = 0 \)

The cosine function is equal to zero at odd multiples of \( \frac{\pi}{2} \):
\[ \frac{x}{5} = \frac{(2m+1)\pi}{2} \quad \text{for } m \in \mathbb{Z} \]
To find \( x \), we rearrange the equation:
\[ x = 5 \cdot \frac{(2m+1)\pi}{2} = \frac{5(2m+1)\pi}{2} \]

### Step 3: Equate the two expressions for \( x \)

We have two expressions for \( x \):
1. \( x = \frac{5n\pi}{6} \)
2. \( x = \frac{5(2m+1)\pi}{2} \)

Setting these equal gives:
\[ \frac{5n\pi}{6} = \frac{5(2m+1)\pi}{2} \]

### Step 4: Simplify the equation

Dividing both sides by \( 5\pi \) (assuming \( \pi \neq 0 \)):
\[ \frac{n}{6} = \frac{2m+1}{2} \]
Cross-multiplying yields:
\[ 2n = 6(2m + 1) \]
\[ 2n = 12m + 6 \]
\[ n = 6m + 3 \]

### Step 5: General solution for \( x \)

Substituting \( n \) back into the equation for \( x \):
\[ x = \frac{5(6m + 3)\pi}{6} = 5(m + \frac{1}{2})\pi \]

Thus, the general solution for \( x \) can be expressed as:
\[ x = 5\left(n + \frac{1}{2}\right)\pi \quad \text{for } n \in \mathbb{Z} \]

### Conclusion

The solutions for the equations \( \sin\left(\frac{6}{5}x\right) = 0 \) and \( \cos\left(\frac{x}{5}\right) = 0 \) are given by:
\[ x = \frac{5n\pi}{6} \quad \text{and} \quad x = \frac{5(2m+1)\pi}{2} \]