Circular & Periodic Functions

Cosine is x, sine is y

Because the radius is $1$, the right-triangle definitions collapse into coordinates. For an angle $\theta$, the hypotenuse of the reference triangle is the radius, which is $1$ — so the adjacent side is just $\cos\theta$ and the opposite side is just $\sin\theta$. The point on the circle is $(\cos\theta, \sin\theta)$.

This makes the axis angles free. At $0^\circ$ the point is $(1, 0)$, so $\cos 0^\circ = 1$ and $\sin 0^\circ = 0$. At $90^\circ$ the point is $(0, 1)$; at $180^\circ$ it's $(-1, 0)$; at $270^\circ$ it's $(0, -1)$. No triangle needed — just read the coordinates.

Tangent comes along for the ride: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, the $y$-coordinate divided by the $x$-coordinate. Wherever cosine is $0$ (at $90^\circ$ and $270^\circ$), tangent is undefined.

The diagram below shows the setup: the radius to the point on the circle is the angle's terminal side, and the point's coordinates are $(\cos\theta, \sin\theta)$ — cosine measured horizontally, sine measured vertically.

The three points worth memorizing

In the first quadrant there are three special angles, and their coordinates use only two values: $\dfrac{1}{2}$, $\dfrac{\sqrt{2}}{2}$, and $\dfrac{\sqrt{3}}{2}$. At $30^\circ$ the point is $\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$. At $45^\circ$ it's $\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$. At $60^\circ$ it's $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$.

A pattern makes these easy to keep straight: as the angle grows from $30^\circ$ to $60^\circ$, the point climbs the circle — so $x$ shrinks and $y$ grows. The small value $\dfrac{1}{2}$ and the big value $\dfrac{\sqrt{3}}{2}$ simply trade places, and $45^\circ$ is the balanced middle where both coordinates match.

In radians, these angles are $\dfrac{\pi}{6}$, $\dfrac{\pi}{4}$, and $\dfrac{\pi}{3}$. Same points, same values — radians are just a different unit for measuring the rotation, where a full circle is $2\pi$ instead of $360^\circ$.

Reference angles and signs in the other quadrants

Every angle outside the first quadrant borrows its values from a first-quadrant twin called the reference angle — the acute angle between the terminal side and the $x$-axis. For $150^\circ$ the reference angle is $180^\circ - 150^\circ = 30^\circ$; for $225^\circ$ it's $225^\circ - 180^\circ = 45^\circ$; for $300^\circ$ it's $360^\circ - 300^\circ = 60^\circ$.

The reference angle supplies the size of the sine and cosine; the quadrant supplies the signs. In Quadrant II, $x$ is negative and $y$ is positive, so cosine is negative and sine is positive. In Quadrant III both coordinates are negative, so both sine and cosine are negative. In Quadrant IV, $x$ is positive and $y$ is negative, so cosine is positive and sine is negative.

So evaluating something like $\cos 225^\circ$ is a two-part job: reference angle $45^\circ$ gives the size $\dfrac{\sqrt{2}}{2}$, and Quadrant III makes it negative — $\cos 225^\circ = -\dfrac{\sqrt{2}}{2}$.

Worked examples

Try one yourself

Common questions