Find the value of `sin 2295^@`.

Answer

To find the value of \( \sin 2295^\circ \), we can follow these steps:

### Step 1: Reduce the angle
First, we need to reduce \( 2295^\circ \) to an equivalent angle within the range of \( 0^\circ \) to \( 360^\circ \). We can do this by subtracting multiples of \( 360^\circ \).

\[ 2295^\circ = 6 \times 360^\circ + 135^\circ \]

### Step 2: Use the sine periodicity property
According to the sine function's periodicity, we have:

\[ \sin(n \times 360^\circ + \theta) = \sin(\theta) \]

In our case, \( n = 6 \) and \( \theta = 135^\circ \). Therefore:

\[ \sin(2295^\circ) = \sin(135^\circ) \]

### Step 3: Calculate \( \sin(135^\circ) \)
Next, we need to find the value of \( \sin(135^\circ) \). We know that:

\[ 135^\circ = 180^\circ - 45^\circ \]

Using the sine identity for angles in the second quadrant:

\[ \sin(180^\circ - \theta) = \sin(\theta) \]

Thus:

\[ \sin(135^\circ) = \sin(45^\circ) \]

### Step 4: Find \( \sin(45^\circ) \)
The value of \( \sin(45^\circ) \) is known to be:

\[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \]

### Final Result
Therefore, we conclude that:

\[ \sin(2295^\circ) = \sin(135^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \]

### Summary
The value of \( \sin(2295^\circ) \) is \( \frac{\sqrt{2}}{2} \).

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