angle between two planes :
2x + y - 2z = 5 and 3x - 6y - 2z = 7 is `sin^(-1) ((4)/(21))`

Answer

To find the angle between two planes given by the equations \(2x + y - 2z = 5\) and \(3x - 6y - 2z = 7\), we can follow these steps:

### Step 1: Identify the normal vectors of the planes
The coefficients of \(x\), \(y\), and \(z\) in the plane equations represent the components of the normal vectors.

For the first plane \(2x + y - 2z = 5\):
- Normal vector \(n_1 = \langle 2, 1, -2 \rangle\)

For the second plane \(3x - 6y - 2z = 7\):
- Normal vector \(n_2 = \langle 3, -6, -2 \rangle\)

### Step 2: Calculate the dot product of the normal vectors
The dot product \(n_1 \cdot n_2\) is calculated as follows:
\[ n_1 \cdot n_2 = (2)(3) + (1)(-6) + (-2)(-2) = 6 - 6 + 4 = 4 \]

### Step 3: Calculate the magnitudes of the normal vectors
The magnitude of \(n_1\) is:
\[ |n_1| = \sqrt{2^2 + 1^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

The magnitude of \(n_2\) is:
\[ |n_2| = \sqrt{3^2 + (-6)^2 + (-2)^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

### Step 4: Use the dot product to find the cosine of the angle
Using the formula for the cosine of the angle \(\theta\) between the two normal vectors:
\[ \cos \theta = \frac{n_1 \cdot n_2}{|n_1| |n_2|} = \frac{4}{3 \cdot 7} = \frac{4}{21} \]

### Step 5: Find the angle using the inverse cosine
To find the angle \(\theta\), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{4}{21}\right) \]

### Step 6: Relate to sine
Since the problem states that the angle is given as \(\sin^{-1}\left(\frac{4}{21}\right)\), we need to check if this is correct. We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, if \(\cos \theta = \frac{4}{21}\), we can find \(\sin \theta\):
\[ \sin^2 \theta = 1 - \left(\frac{4}{21}\right)^2 = 1 - \frac{16}{441} = \frac{441 - 16}{441} = \frac{425}{441} \]
\[ \sin \theta = \sqrt{\frac{425}{441}} = \frac{\sqrt{425}}{21} \]

### Conclusion
The angle between the two planes is given by \(\cos^{-1}\left(\frac{4}{21}\right)\), not \(\sin^{-1}\left(\frac{4}{21}\right)\). Therefore, the statement in the question is **false**.