$$ \text{simplify}\colon\text{ }\sin $$$$ (2460^{\doubleprime}) $$

To simplify the expression $\sin(2460^{\circ})$, we can use the periodicity of the sine function. The sine function has a period of $360^\circ$, which means:

\[
\sin(x) = \sin(x + 360^\circ k)
\]
for any integer $k$.

### Step 1: Reduce the angle modulo $360^\circ$
We can find an equivalent angle within the standard range $[0^\circ, 360^\circ)$ by subtracting multiples of $360^\circ$ from $2460^\circ$.

\[
2460^\circ \div 360^\circ = 6.8333
\]
This tells us that $2460^\circ$ is equivalent to $2460^\circ - 6 \times 360^\circ$:

\[
2460^\circ - 2160^\circ = 300^\circ
\]

### Step 2: Simplify using the reduced angle
So, $\sin(2460^\circ)$ simplifies to:

\[
\sin(2460^\circ) = \sin(300^\circ)
\]

### Step 3: Evaluate $\sin(300^\circ)$
The angle $300^\circ$ is in the fourth quadrant, where the sine function is negative. Specifically:

\[
\sin(300^\circ) = -\sin(360^\circ - 300^\circ) = -\sin(60^\circ)
\]

Since $\sin(60^\circ) = \frac{\sqrt{3}}{2}$:

\[
\sin(300^\circ) = -\frac{\sqrt{3}}{2}
\]

### Final Answer
Thus, $\sin(2460^\circ) = -\frac{\sqrt{3}}{2}$.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the value of $\cos(2460^\circ)$?
2. How does the cosine function simplify for angles like $2460^\circ$?
3. What is the general formula for simplifying $\sin(x)$ where $x$ is an arbitrary large angle?
4. Can you simplify $\tan(2460^\circ)$?
5. How would you use the unit circle to visualize $\sin(2460^\circ)$?

**Tip:** Always reduce large angles modulo $360^\circ$ to find their equivalent acute or reference angle, making trigonometric calculations easier.

Discover how to simplify sin(2460 degrees) using trigonometric identities and periodicity of the sine function. Step-by-step solution provided with detailed calculations and angle reduction techniques. Suitable for advanced high school students studying trigonometry.

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