Evaluate cos(225 degrees), giving an exact value.

Step-by-step solution

1. 1

   Locate the angle's quadrant. $180^\circ<225^\circ<270^\circ$, so the terminal side lies in the third quadrant, where both coordinates on the unit circle are negative. Cosine reads the $x$-coordinate, so the answer must be negative.

2. 2

   Find the reference angle. In the third quadrant the reference angle is the angle past $180^\circ$:

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

3. 3

   Apply the third-quadrant identity. Equivalently $\cos(180^\circ+\theta)=-\cos\theta$:

   $\cos 225^\circ=\cos(180^\circ+45^\circ)=-\cos 45^\circ$

4. 4

   Substitute the special-angle value.

   $\cos 45^\circ=\frac{\sqrt2}{2}\quad\Longrightarrow\quad \cos 225^\circ=-\frac{\sqrt2}{2}$

5. 5

   Check numerically and against the sign rule. $-\tfrac{\sqrt2}{2}\approx-0.7071068$, and a direct evaluation of $\cos 225^\circ$ gives $-0.7071068$ $\checkmark$. The negative sign matches the third-quadrant prediction, and $225^\circ$ in radians is $\tfrac{5\pi}{4}$, the point $\left(-\tfrac{\sqrt2}{2},-\tfrac{\sqrt2}{2}\right)$ on the unit circle.