When the axes are rotated through an angle`pi/2`, find the new coordinates of the point` (alpha,0)`

Answer

To find the new coordinates of the point \((\alpha, 0)\) after rotating the axes through an angle of \(\frac{\pi}{2}\), we can follow these steps:

### Step 1: Identify the rotation angle and the original coordinates
The original coordinates of the point are given as \((\alpha, 0)\). The angle of rotation is \(\theta = \frac{\pi}{2}\).

### Step 2: Use the rotation formulas
The formulas for the new coordinates \((x', y')\) after rotation are:
\[ x' = x \cos \theta + y \sin \theta \]
\[ y' = -x \sin \theta + y \cos \theta \]

### Step 3: Substitute the values
Substituting \(x = \alpha\) and \(y = 0\) into the formulas, we get:
\[ x' = \alpha \cos\left(\frac{\pi}{2}\right) + 0 \cdot \sin\left(\frac{\pi}{2}\right) \]
\[ y' = -\alpha \sin\left(\frac{\pi}{2}\right) + 0 \cdot \cos\left(\frac{\pi}{2}\right) \]

### Step 4: Calculate \(\cos\) and \(\sin\) values
We know that:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \sin\left(\frac{\pi}{2}\right) = 1 \]

### Step 5: Simplify the expressions
Now we can simplify the expressions for \(x'\) and \(y'\):
\[ x' = \alpha \cdot 0 + 0 \cdot 1 = 0 + 0 = 0 \]
\[ y' = -\alpha \cdot 1 + 0 \cdot 0 = -\alpha + 0 = -\alpha \]

### Step 6: Final result
Thus, the new coordinates after the rotation are:
\[ (x', y') = (0, -\alpha) \]

### Summary of the solution
The new coordinates of the point \((\alpha, 0)\) after rotating the axes through an angle of \(\frac{\pi}{2}\) are \((0, -\alpha)\).

---