To find the exact value of $\sin \left( -\frac{3\pi}{2} \right)$ using the even-odd properties of trigonometric functions, let's break down the solution:

Step 1: Recognize the Even-Odd Property for Sine

The sine function is an odd function, meaning that: $\sin(-\theta) = -\sin(\theta)$ This property applies here because we have a negative angle, $-\frac{3\pi}{2}$.

Step 2: Apply the Property

Using the odd property of sine, we get: $\sin \left( -\frac{3\pi}{2} \right) = -\sin \left( \frac{3\pi}{2} \right)$

Step 3: Evaluate $\sin \left( \frac{3\pi}{2} \right)$

Now, we need to find $\sin \left( \frac{3\pi}{2} \right)$. This angle corresponds to $\frac{3\pi}{2}$ radians, which is 270° on the unit circle. The sine of 270° (or $\frac{3\pi}{2}$ radians) is: $\sin \left( \frac{3\pi}{2} \right) = -1$

Step 4: Substitute Back

Now we substitute back into our expression: $\sin \left( -\frac{3\pi}{2} \right) = -(-1) = 1$

Conclusion

The exact value of $\sin \left( -\frac{3\pi}{2} \right)$ is $1$.

Answer Choices

• The correct even-odd property to use is option C: $\sin(-\theta) = -\sin(\theta)$.
• The correct answer for $\sin \left( -\frac{3\pi}{2} \right)$ is option A with the blank filled as $1$.

Would you like further clarification on this process, or have additional questions? Here are some related questions for deeper exploration:

1. What are the even-odd properties for other trigonometric functions like cosine and tangent?
2. How can understanding the unit circle help in evaluating trigonometric functions at key angles?
3. Why is the sine function considered an odd function, and how does this affect its graph?
4. What are the values of sine and cosine for other multiples of $\pi$ and $\frac{\pi}{2}$?
5. How do even-odd properties apply to trigonometric identities in general?

Tip: Knowing the unit circle and key angles like $\frac{\pi}{2}, \pi, \frac{3\pi}{2},$ and $2\pi$ can help in quickly evaluating trigonometric functions without a calculator.