Unit Circle - Essential Trigonometric Values

Related Pairs for Sine and Cosine

Defined by the equation $x^{2}+y^{2}\,=\,1$, the unit circle is the collection of points that lie one unit from the origin. For trigonometry, it relates directions, called out in degrees or radians, to their cosine and sine. We measure angles counterclockwise from the positive $x$-axis, shown as $0^{\circ }$ or $0$ radians, and each ordered pair, such as $(1,0)$, is both a point on the unit circle and the cosine and sine in that direction or angle. In the image, we have color coded points and directions which are related through absolute values.

Each red point of the unit circle lies on an axis, and has one coordinate of $0$, and the other of absolute value $1$. These are all a multiple of $\pi /2$.

On the other hand, each point in a diagonal direction (shown in green) has a coordinate pair which are both ${\sqrt {2}}/2$ in absolute value. These are all an odd multiple of $\pi /4$.

Finally, each blue direction has coordinates with two different absolute values, ${\sqrt {3}}/2$  and $1/2$. These fill out the remaining multiples of $\pi /6$, or those which are not already shown in red.

It is important to realize that ${\sqrt {3}}/2\,>\,1/2$, so whichever direction ($x$ or $y$) seems smaller will take $1/2$  as the absolute value of its coordinate.

As an example, consider the angle/direction $\pi /6$. If we first consider the green diagonal at $\pi /4$, where both coordinates share the same absolute value, we can see that the point on the unit circle at $\pi /6$ is to the right and below the diagonal. Thus, its $x$ coordinate is bigger than its $y$ coordinate, as can be seen in the image.

Also, recall that adding or subtracting the measure of a circle - either $360^{\circ }$ or $2\pi$ - does NOT change the direction we are heading. As such, the angles $-2\pi ,\,0,\,2\pi$ and $4\pi$ all point in the direction of the positive $x$-axis. We may call such angles coterminal.