To solve the problem step by step, let's start by interpreting what is asked:

Problem:

You are asked to find the exact value of $\sin(225^\circ)$ using a reference angle.

Step 1: Determine the Reference Angle for $225^\circ$

1. Reference Angle: The reference angle is the smallest angle the terminal side of $225^\circ$ makes with the x-axis. To find the reference angle for an angle in the third quadrant (which $225^\circ$ is), we subtract $180^\circ$ from $225^\circ$:

$\text{Reference angle} = 225^\circ - 180^\circ = 45^\circ$

So, the reference angle is $45^\circ$.

Step 2: Find the Exact Value of $\sin(225^\circ)$

Since $225^\circ$ lies in the third quadrant, sine is negative in this quadrant. The sine of the reference angle $45^\circ$ is:

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

Thus, for $225^\circ$ (in the third quadrant), we have:

$\sin(225^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}$

Final Answer:

$\sin(225^\circ) = -\frac{\sqrt{2}}{2}$

So, the correct choice is:

• $A.$ $\sin(225^\circ) = -\frac{\sqrt{2}}{2}$

Let me know if you would like more details, or if you have any questions! Here are some related questions to expand on this topic:

1. What are reference angles and how do they apply to trigonometric functions?
2. How do you determine the signs of trigonometric functions in different quadrants?
3. What is the exact value of $\cos(225^\circ)$?
4. How do you calculate the sine of angles like $135^\circ$ or $315^\circ$?
5. Can you find $\sin(225^\circ)$ using the unit circle?

Tip:

Always remember that the sign of a trigonometric function depends on the quadrant where the angle lies. Sine is positive in the first and second quadrants, and negative in the third and fourth quadrants.