The vector makes angles `Sin^-1(sqrt(2146)/65) and Sin^-1(12/13)` with the `X` and `Y` axes . Find the Sin ratio of the angle that is makes with the `Z`-axis.

Options

A. `3/5`
B. `4/5`
C. `5/6`
D. `7/8`

Explanation

To find the sine ratio of the angle that a vector makes with the Z-axis, given the angles it makes with the X and Y axes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the angles**:
Let \( \alpha \) be the angle between the vector and the X-axis, and \( \beta \) be the angle between the vector and the Y-axis. We are given:
\[ \alpha = \sin^{-1}\left(\frac{\sqrt{2146}}{65}\right) \]
\[ \beta = \sin^{-1}\left(\frac{12}{13}\right) \]

2. **Find the sine values**:
From the definitions of sine, we have:
\[ \sin \alpha = \frac{\sqrt{2146}}{65} \]
\[ \sin \beta = \frac{12}{13} \]

3. **Use the relationship for cosines**:
We know that:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
where \( \gamma \) is the angle with the Z-axis. We can find \( \cos \alpha \) and \( \cos \beta \) using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \).

4. **Calculate \( \cos^2 \alpha \)**:
\[ \cos^2 \alpha = 1 - \sin^2 \alpha = 1 - \left(\frac{\sqrt{2146}}{65}\right)^2 = 1 - \frac{2146}{4225} \]
\[ \cos^2 \alpha = \frac{4225 - 2146}{4225} = \frac{2079}{4225} \]

5. **Calculate \( \cos^2 \beta \)**:
\[ \cos^2 \beta = 1 - \sin^2 \beta = 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} \]
\[ \cos^2 \beta = \frac{169 - 144}{169} = \frac{25}{169} \]

6. **Substitute into the cosine relationship**:
Now substitute \( \cos^2 \alpha \) and \( \cos^2 \beta \) into the equation:
\[ \frac{2079}{4225} + \frac{25}{169} + \cos^2 \gamma = 1 \]

7. **Find a common denominator**:
The common denominator for \( 4225 \) and \( 169 \) is \( 4225 \times 169 \). Convert each term:
\[ \frac{2079 \cdot 169}{4225 \cdot 169} + \frac{25 \cdot 4225}{169 \cdot 4225} + \cos^2 \gamma = 1 \]
\[ \frac{351171 + 105625}{4225 \cdot 169} + \cos^2 \gamma = 1 \]
\[ \frac{456796}{4225 \cdot 169} + \cos^2 \gamma = 1 \]

8. **Solve for \( \cos^2 \gamma \)**:
Rearranging gives:
\[ \cos^2 \gamma = 1 - \frac{456796}{4225 \cdot 169} \]
To simplify, calculate \( 1 \) in terms of the common denominator:
\[ \cos^2 \gamma = \frac{4225 \cdot 169 - 456796}{4225 \cdot 169} \]

9. **Calculate the numerator**:
Calculate \( 4225 \cdot 169 \):
\[ 4225 \cdot 169 = 714425 \]
Now calculate \( 714425 - 456796 = 257629 \):
\[ \cos^2 \gamma = \frac{257629}{714425} \]

10. **Find \( \sin^2 \gamma \)**:
Since \( \sin^2 \gamma + \cos^2 \gamma = 1 \):
\[ \sin^2 \gamma = 1 - \cos^2 \gamma = 1 - \frac{257629}{714425} = \frac{714425 - 257629}{714425} = \frac{456796}{714425} \]

11. **Calculate \( \sin \gamma \)**:
Taking the square root gives:
\[ \sin \gamma = \sqrt{\frac{456796}{714425}} = \frac{\sqrt{456796}}{\sqrt{714425}} \]

12. **Final simplification**:
Calculate the square roots:
\[ \sin \gamma = \frac{676}{845} = \frac{4}{5} \]

### Final Answer:
Thus, the sine ratio of the angle that the vector makes with the Z-axis is:
\[ \sin \gamma = \frac{4}{5} \]