Unit circle

In a right triangle the two acute angles always lie between 0 and 90 degrees, and larger angles never occur there. But what about 120 or 270 degrees? The unit circle extends sine and cosine to any angle at all, and it also makes clear why these values are sometimes negative.

A circle of radius 1

The unit circle is a circle around the origin with radius 1. A point travels along this circle, and the angle $\alpha$ is measured from the positive x-axis counterclockwise. The coordinates of this point are exactly the cosine and the sine of the angle.

x is cosine, y is sine

Because the radius is 1, $\cos\alpha = \frac{x}{1}$ becomes simply $\cos\alpha = x$, and the sine becomes $\sin\alpha = y$. So the x-value of the point is the cosine and the y-value is the sine. This lets you read every value straight off the circle.

Signs in the quadrants

The circle is split into four quarters, counted counterclockwise. In the top right (quadrant I) both x and y are positive, so sine and cosine are too. In the top left (quadrant II) x is negative, so cosine turns negative while sine stays positive. In the bottom left (quadrant III) both are negative, and in the bottom right (quadrant IV) only the sine is negative. That is why $\cos 120° = -0.5$ while $\sin 120°$ stays positive.