Simplify: (A) Sin(2460°) (B) tan(- 150°)

To simplify trigonometric functions for angles, it's important to reduce the angles to their standard form, which lies between \( 0^\circ \) and \( 360^\circ \). Here's how we can simplify each expression:

### (A) Simplify \( \sin(2460^\circ) \)

1. **Reduce \( 2460^\circ \) to an equivalent angle between \( 0^\circ \) and \( 360^\circ \):**

   We can reduce any angle by subtracting multiples of \( 360^\circ \) (since one full rotation is \( 360^\circ \)):

   \[
   2460^\circ \div 360^\circ = 6.8333
   \]

   So, subtract \( 360^\circ \times 6 = 2160^\circ \) from \( 2460^\circ \):

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

   Therefore:

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

2. **Evaluate \( \sin(300^\circ) \):**

   \( 300^\circ \) lies in the fourth quadrant, where the sine function is negative. The reference angle for \( 300^\circ \) is \( 360^\circ - 300^\circ = 60^\circ \).

   Therefore:

   \[
   \sin(300^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2}
   \]

   So:

   \[
   \sin(2460^\circ) = -\frac{\sqrt{3}}{2}
   \]

### (B) Simplify \( \tan(-150^\circ) \)

1. **Reduce \( -150^\circ \) to a positive angle:**

   To convert a negative angle into a positive one, add \( 360^\circ \):

   \[
   -150^\circ + 360^\circ = 210^\circ
   \]

   Therefore:

   \[
   \tan(-150^\circ) = \tan(210^\circ)
   \]

2. **Evaluate \( \tan(210^\circ) \):**

   \( 210^\circ \) lies in the third quadrant, where the tangent function is positive. The reference angle for \( 210^\circ \) is \( 210^\circ - 180^\circ = 30^\circ \).

   Therefore:

   \[
   \tan(210^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}}
   \]

   So:

   \[
   \tan(-150^\circ) = \frac{1}{\sqrt{3}}
   \]

Would you like further details or have any questions?

Here are 5 related questions:
1. Simplify \( \cos(750^\circ) \).
2. Simplify \( \sin(-45^\circ) \).
3. Simplify \( \tan(1020^\circ) \).
4. Find the exact value of \( \cos(330^\circ) \).
5. Simplify \( \cot(210^\circ) \).

**Tip:** For trigonometric functions, always reduce the angle to within \( 0^\circ \) to \( 360^\circ \) first to simplify calculations.

Explore how to simplify trigonometric functions for angles: Sin(2460°) and Tan(-150°). Understand angle reduction techniques, trigonometric identities, and calculations involving sine and tangent functions. Suitable for high school students studying trigonometry.

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