Solution: \( 225^\circ \) lies in the third quadrant, where sine is negative. Reference angle: \( 225^\circ - 180^\circ = 45^\circ \). Thus, \( \sin 225^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2} \). \boxed{-\dfrac{\sqrt{2}}{2}}

["Understanding the Sine of 225°: Location, Sign, and Reference Angle Explained", "When studying trigonometric functions, one crucial concept is how angles in different quadrants affect the sign of sine, cosine, and tangent values. A classic example is the angle ( 225^\circ ), commonly analyzed to illustrate how quadrants determine function signs.", "### Where is ( 225^\circ ) Located?", "The angle ( 225^\circ ) lies in the third quadrant of the coordinate plane. Recall that the full circle is divided into four quadrants:", "- First quadrant: ( 0^\circ ) to ( 90^\circ ) — all trigonometric functions are positive.\n- Second quadrant: ( 90^\circ ) to ( 180^\circ ) — sine is positive, cosine and tangent are negative.\n- Third quadrant: ( 180^\circ ) to ( 270^\circ ) — both sine and cosine are negative; tangent is positive.\n- Fourth quadrant: ( 270^\circ ) to ( 360^\circ ) — cosine is positive, sine is negative.", "Since ( 225^\circ ) is between ( 180^\circ ) and ( 270^\circ ), it clearly falls in the third quadrant.", "### Determining the Sign of Sine in the Third Quadrant", "In the third quadrant, sine values are negative. This behavior counterintuitively arises because sine corresponds to the y-coordinate on the unit circle, and y-values are negative between ( 180^\circ ) and ( 270^\circ ).", "### Finding the Reference Angle for ( 225^\circ )", "To compute trigonometric functions of angles like ( 225^\circ ), mathematicians use the reference angle—the acute angle between the terminal side of the given angle and the x-axis.", "To find the reference angle for ( 225^\circ ):\n[\n\ ext{Reference angle} = 225^\circ - 180^\circ = 45^\circ\n]", "This means the terminal side of ( 225^\circ ) makes a ( 45^\circ ) angle with the negative x-axis in the third quadrant.", "### Calculating ( \sin 225^\circ )", "Because ( 45^\circ ) is a standard angle with known sine value:\n[\n\sin 45^\circ = \frac{\sqrt{2}}{2}\n]", "But since ( 225^\circ ) is in the third quadrant, where sine is negative, we take the negative of this value:\n[\n\sin 225^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2}\n]", "### Conclusion", "So, the exact value of ( \sin 225^\circ ) is ( -\frac{\sqrt{2}}{2} ). Understanding reference angles and quadrant signs is essential for accurately evaluating trigonometric functions. Whether you're solving equations, graphing, or real-world applications, recognizing that ( 225^\circ ) lies in the third quadrant — where sine is negative — helps build strong foundations in trigonometry.", "[\n\boxed{-\dfrac{\sqrt{2}}{2}}\n]"]