Solve `sin x =-1 ` for principal solution as well as general solution:
(i) `x in ` { angle with measures given in radians}
(ii) `x in` {angle with measures given in degres}.

Answer

To solve the equation \( \sin x = -1 \), we will find both the principal solution and the general solution in radians and degrees.

### Step 1: Principal Solution in Radians
We start with the equation:
\[ \sin x = -1 \]
We know that the sine function achieves the value of -1 at specific points on the unit circle. The principal solution, which lies in the interval \( [0, 2\pi) \), occurs at:
\[ x = \frac{3\pi}{2} \]

### Step 2: General Solution in Radians
The general solution for the sine function can be expressed as:
\[ x = n\pi + (-1)^n \cdot \alpha \]
where \( \alpha \) is the angle at which the sine function equals -1. From Step 1, we have \( \alpha = \frac{3\pi}{2} \). Thus, the general solution becomes:
\[ x = n\pi + (-1)^n \cdot \frac{3\pi}{2} \]
This can be simplified to:
\[ x = n\pi + \frac{3\pi}{2} \quad \text{for even } n \]
\[ x = n\pi - \frac{3\pi}{2} \quad \text{for odd } n \]

### Step 3: Principal Solution in Degrees
Next, we convert the principal solution found in radians to degrees. We know that:
\[ \frac{3\pi}{2} \text{ radians} = 270^\circ \]
Thus, the principal solution in degrees is:
\[ x = 270^\circ \]

### Step 4: General Solution in Degrees
For the general solution in degrees, we can use the same formula as before, but convert \( \pi \) to degrees:
\[ \pi \text{ radians} = 180^\circ \]
Thus, the general solution in degrees is:
\[ x = n \cdot 180^\circ + (-1)^n \cdot 270^\circ \]
This can also be expressed as:
\[ x = 180n + 270 \quad \text{for even } n \]
\[ x = 180n - 270 \quad \text{for odd } n \]

### Summary of Solutions
1. **Principal Solution in Radians**: \( x = \frac{3\pi}{2} \)
2. **General Solution in Radians**: \( x = n\pi + (-1)^n \cdot \frac{3\pi}{2} \)
3. **Principal Solution in Degrees**: \( x = 270^\circ \)
4. **General Solution in Degrees**: \( x = 180n + (-1)^n \cdot 270^\circ \)