Find the exact value of cosecant, 240, degreescsc240
∘
in simplest form with a rational denominator.

To find the exact value of \(\csc 240^\circ\), we need to understand the relationship between the cosecant function and the sine function. Recall that:

\[
\csc \theta = \frac{1}{\sin \theta}
\]

Thus, finding \(\csc 240^\circ\) requires us first to determine \(\sin 240^\circ\).

### Step 1: Determine the reference angle and quadrant

1. **Identify the quadrant:**  
   The angle \(240^\circ\) is in the third quadrant since it is between \(180^\circ\) and \(270^\circ\).

2. **Find the reference angle:**  
   The reference angle for an angle in the third quadrant is given by:

   \[
   240^\circ - 180^\circ = 60^\circ
   \]

### Step 2: Evaluate \(\sin 240^\circ\)

In the third quadrant, the sine function is negative. The sine of the reference angle \(60^\circ\) is:

\[
\sin 60^\circ = \frac{\sqrt{3}}{2}
\]

Thus, in the third quadrant:

\[
\sin 240^\circ = -\frac{\sqrt{3}}{2}
\]

### Step 3: Calculate \(\csc 240^\circ\)

Now, using the reciprocal identity for cosecant:

\[
\csc 240^\circ = \frac{1}{\sin 240^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}}
\]

To simplify, we multiply the numerator and the denominator by 2:

\[
\csc 240^\circ = \frac{2}{-\sqrt{3}}
\]

To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\):

\[
\csc 240^\circ = \frac{2 \cdot \sqrt{3}}{-\sqrt{3} \cdot \sqrt{3}} = \frac{2\sqrt{3}}{-3} = -\frac{2\sqrt{3}}{3}
\]

### Conclusion

The exact value of \(\csc 240^\circ\) in its simplest form with a rational denominator is:

\[
\csc 240^\circ = -\frac{2\sqrt{3}}{3}
\]

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What is the value of \(\csc 300^\circ\)?
2. How do you find the reference angle for an angle in the fourth quadrant?
3. What are the values of sine and cosine for common angles like \(30^\circ, 45^\circ,\) and \(60^\circ\)?
4. How do you rationalize a denominator?
5. What is the general method for finding the cosecant of any angle?

**Tip:** When working with trigonometric functions, always consider the quadrant to determine the sign of the function value.

Learn how to find the exact value of \(\csc 240^\circ\) in simplest form with a rational denominator. This problem involves trigonometric functions and reciprocal identities, specifically focusing on \(\sin 240^\circ\) and its relation to \(\csc 240^\circ\). Suitable for Grades 10-12.

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