Exact sine and cosine values for unit circle angles

Key concept: This question checks unit circle symmetry, reference angles, and exact trigonometric values. The angles are special, so the answers should be found by rotation and quadrant signs rather than approximation.

Step by step

Step 1: Use reference angles and rotation

On the unit circle, the coordinates of a point at angle $\theta$ are $(\cos\theta,\sin\theta)$. That means once the angle is located on the circle, the cosine is the $x$-coordinate and the sine is the $y$-coordinate.

For $-225^\circ$, add $360^\circ$ to find a coterminal angle:

$-225^\circ+360^\circ=135^\circ$

So $-225^\circ$ and $135^\circ$ land on the same point.

Step 2: Find the exact values at 135°

The angle $135^\circ$ is in Quadrant II with reference angle $45^\circ$. The exact unit circle values for $45^\circ$ are

$\sin 45^\circ=\frac{\sqrt{2}}{2}, \qquad \cos 45^\circ=\frac{\sqrt{2}}{2}$

In Quadrant II, sine is positive and cosine is negative. Therefore:

$\sin 135^\circ=\frac{\sqrt{2}}{2}, \qquad \cos 135^\circ=-\frac{\sqrt{2}}{2}$

Since $-225^\circ$ is coterminal with $135^\circ$, it has the same exact values:

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

Step 3: Key unit circle idea

The important point is that trigonometric values depend on position on the unit circle, not on whether the angle is written as positive or negative. Coterminal angles always produce the same sine and cosine values.

When a problem asks for exact values without a calculator, you should immediately check whether the angle is a special angle such as $30^\circ$, $45^\circ$, or $60^\circ$, or a coterminal angle of one of these. That saves time and avoids decimal approximations.

Pitfall alert

Students often treat a negative angle as if it must give negative sine and cosine values. That is not true. The sign depends on the quadrant of the terminal side, not on whether the original angle is negative. Another common mistake is mixing up sine and cosine coordinates on the unit circle. Remember: cosine is the horizontal coordinate, and sine is the vertical coordinate. For $-225^\circ$, the clean method is to convert to a coterminal angle first.

Try different conditions

If the question changed to find $\sin(-45^\circ)$ and $\cos(-45^\circ)$, the answer would be different because $-45^\circ$ lies in Quadrant IV. There, sine is negative and cosine is positive, so $\sin(-45^\circ)=-\frac{\sqrt{2}}{2}$ and $\cos(-45^\circ)=\frac{\sqrt{2}}{2}$. The same unit circle method still applies: locate the terminal side, identify the reference angle, then apply the correct signs for the quadrant.

Further reading

unit circle, coterminal angles, reference angle