Find the exact value of sin(225 degrees).

Step-by-step solution

1. 1

   Locate the angle's quadrant. Since $180^\circ<225^\circ<270^\circ$, the terminal side lies in Quadrant III. There both sine and cosine are negative (only tangent is positive), so the answer must carry a minus sign.

2. 2

   Find the reference angle. For a Quadrant III angle the reference angle is the angle measured back to the negative $x$-axis:

   $225^\circ-180^\circ=45^\circ$

   A reference angle of $45^\circ$ means the value will be one of the exact special values, so no calculator is needed.

3. 3

   Recall the special value. From the $45^\circ$-$45^\circ$-$90^\circ$ triangle with legs $1,1$ and hypotenuse $\sqrt2$:

   $\sin45^\circ=\frac{1}{\sqrt2}=\frac{\sqrt2}{2}$

4. 4

   Attach the quadrant sign. The magnitude comes from the reference angle and the sign from the quadrant:

   $\sin225^\circ=-\sin45^\circ=-\frac{\sqrt2}{2}$

5. 5

   Cross-check with the unit circle coordinates. The point at $225^\circ$ is $\left(-\tfrac{\sqrt2}{2},-\tfrac{\sqrt2}{2}\right)$, and sine is the $y$-coordinate — negative, as found ✓. Numerically $-\tfrac{\sqrt2}{2}=-0.70711$, and evaluating $\sin(225^\circ)$ directly gives $-0.7071068$ ✓, matching to seven decimals.

6. 6

   Note the related values. By the same reference angle, $\cos225^\circ=-\tfrac{\sqrt2}{2}$ and $\tan225^\circ=+1$ — the two negatives cancel in the ratio, which is why tangent is positive in Quadrant III.