Unit Circle

Understanding the Unit Circle

The unit circle is a specific circle that makes trigonometry with angles especially simple and organized. In this chapter, we focus on what the unit circle is, how it is placed on the coordinate plane, and how it helps describe sine, cosine, and tangent values for important angles.

You will use both degrees and radians here, but detailed conversion and definitions of degrees and radians are handled in their own chapters.

The Unit Circle Itself

A unit circle is a circle of radius 1 centered at the origin of the coordinate plane.

In equation form, the unit circle is
$$
x^2 + y^2 = 1.
$$

Any point $(x,y)$ on this circle is exactly distance 1 from the origin.

We will always imagine:

• the center at $(0,0)$,
• radius $1$,
• angles measured from the positive $x$-axis,
• positive angles turning counterclockwise.

Angles and Points on the Unit Circle

When you choose an angle $\theta$ (in degrees or radians), you can draw the terminal side of the angle starting from the origin. Where this ray meets the unit circle, you get a point $(x,y)$ on the circle.

On the unit circle, this point is:

• $x = \cos \theta$,
• $y = \sin \theta$.

So the coordinates of the point on the unit circle corresponding to angle $\theta$ are
$$
(\cos \theta,\ \sin \theta).
$$

This simple fact is one of the main reasons the unit circle is so useful.

Why $(\cos\theta, \sin\theta)$?

Imagine a right triangle formed by:

• the radius from the origin to the point on the circle,
• a vertical line down (or up) to the $x$-axis,
• the $x$-axis between that foot and the origin.

Because the radius length is 1:

• the horizontal leg has length $|\cos\theta|$,
• the vertical leg has length $|\sin\theta|$,

and by choosing appropriate signs based on the direction (left/right, up/down), these become the $x$ and $y$ coordinates.

The Pythagorean identity
$$
\cos^2\theta + \sin^2\theta = 1
$$
matches the equation $x^2 + y^2 = 1$ of the unit circle.

Quadrants and Sign Patterns

The unit circle sits on the coordinate plane, which is divided into four quadrants:

• Quadrant I: $x > 0$, $y > 0$
• Quadrant II: $x < 0$, $y > 0$
• Quadrant III: $x < 0$, $y < 0$
• Quadrant IV: $x > 0$, $y < 0$

Because a point on the unit circle has coordinates $(\cos\theta,\ \sin\theta)$, you can read off the signs of cosine and sine from the quadrant:

• Quadrant I: $\cos\theta > 0$, $\sin\theta > 0$
• Quadrant II: $\cos\theta < 0$, $\sin\theta > 0$
• Quadrant III: $\cos\theta < 0$, $\sin\theta < 0$
• Quadrant IV: $\cos\theta > 0$, $\sin\theta < 0$

Tangent, defined as
$$
\tan\theta = \frac{\sin\theta}{\cos\theta},
$$
follows from these signs. For example, in Quadrant II, sine is positive and cosine is negative, so tangent is negative.

Special Angles and Exact Values

Some angles appear so often that their sine and cosine values are worth memorizing. On the unit circle, these correspond to points with simple coordinate values involving $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, and $\frac{\sqrt{3}}{2}$.

We will list these angles in both degrees and radians.

Axis Angles

These are angles where the terminal side lies directly on one of the axes.

• $0^\circ$ or $0$ radians:
• Point: $(1,0)$
• $\cos 0 = 1$, $\sin 0 = 0$
• $90^\circ$ or $\frac{\pi}{2}$:
• Point: $(0,1)$
• $\cos \frac{\pi}{2} = 0$, $\sin \frac{\pi}{2} = 1$
• $180^\circ$ or $\pi$:
• Point: $(-1,0)$
• $\cos \pi = -1$, $\sin \pi = 0$
• $270^\circ$ or $\frac{3\pi}{2}$:
• Point: $(0,-1)$
• $\cos \frac{3\pi}{2} = 0$, $\sin \frac{3\pi}{2} = -1$
• $360^\circ$ or $2\pi$ (same as $0$):
• Point: $(1,0)$ again.

$30^\circ$ Family ($\pi/6$)

Start with $30^\circ$, or $\theta = \frac{\pi}{6}$, in Quadrant I:

• $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$,
• $\sin \frac{\pi}{6} = \frac{1}{2}$.

Rotating this angle into other quadrants (keeping track of signs) gives related points:

• $150^\circ$ ($\frac{5\pi}{6}$): Quadrant II
• $\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$,
• $\sin \frac{5\pi}{6} = \frac{1}{2}$.
• $210^\circ$ ($\frac{7\pi}{6}$): Quadrant III
• $\cos \frac{7\pi}{6} = -\frac{\sqrt{3}}{2}$,
• $\sin \frac{7\pi}{6} = -\frac{1}{2}$.
• $330^\circ$ ($\frac{11\pi}{6}$): Quadrant IV
• $\cos \frac{11\pi}{6} = \frac{\sqrt{3}}{2}$,
• $\sin \frac{11\pi}{6} = -\frac{1}{2}$.

$45^\circ$ Family ($\pi/4$)

For $45^\circ$ or $\theta = \frac{\pi}{4}$:

• $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$,
• $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$.

By symmetry:

• $135^\circ$ ($\frac{3\pi}{4}$): Quadrant II
• $\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$,
• $\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}$.
• $225^\circ$ ($\frac{5\pi}{4}$): Quadrant III
• $\cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$,
• $\sin \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$.
• $315^\circ$ ($\frac{7\pi}{4}$): Quadrant IV
• $\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}$,
• $\sin \frac{7\pi}{4} = -\frac{\sqrt{2}}{2}$.

$60^\circ$ Family ($\pi/3$)

For $60^\circ$ or $\theta = \frac{\pi}{3}$:

• $\cos \frac{\pi}{3} = \frac{1}{2}$,
• $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$.

Rotating:

• $120^\circ$ ($\frac{2\pi}{3}$): Quadrant II
• $\cos \frac{2\pi}{3} = -\frac{1}{2}$,
• $\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}$.
• $240^\circ$ ($\frac{4\pi}{3}$): Quadrant III
• $\cos \frac{4\pi}{3} = -\frac{1}{2}$,
• $\sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}$.
• $300^\circ$ ($\frac{5\pi}{3}$): Quadrant IV
• $\cos \frac{5\pi}{3} = \frac{1}{2}$,
• $\sin \frac{5\pi}{3} = -\frac{\sqrt{3}}{2}$.

Tangent Values at Special Angles

Using $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$:

• At $0^\circ$ ($0$): $\tan 0 = 0$.
• At $30^\circ$ ($\frac{\pi}{6}$): $\tan \frac{\pi}{6} = \dfrac{1/2}{\sqrt{3}/2} = \dfrac{1}{\sqrt{3}}$.
• At $45^\circ$ ($\frac{\pi}{4}$): $\tan \frac{\pi}{4} = 1$.
• At $60^\circ$ ($\frac{\pi}{3}$): $\tan \frac{\pi}{3} = \dfrac{\sqrt{3}/2}{1/2} = \sqrt{3}$.
• At $90^\circ$ ($\frac{\pi}{2}$): $\tan \frac{\pi}{2}$ is undefined (division by zero, since $\cos \frac{\pi}{2} = 0$).

The same patterns extend to angles in other quadrants, with signs determined by $\sin\theta$ and $\cos\theta$.

Symmetry on the Unit Circle

The unit circle has several useful symmetries that help you find values without memorizing every single one.

Reference Angles

A reference angle is the acute angle (between $0^\circ$ and $90^\circ$) that the terminal side of $\theta$ makes with the $x$-axis.

Angles with the same reference angle share the same absolute values of sine and cosine; only the signs change depending on the quadrant.

For example, $30^\circ$, $150^\circ$, $210^\circ$, and $330^\circ$ all have a reference angle of $30^\circ$. The sine and cosine values are the same up to sign.

Even–Odd Symmetry

From the unit circle you can see:

• $\cos(-\theta) = \cos \theta$ (cosine is an even function),
• $\sin(-\theta) = -\sin \theta$ (sine is an odd function),
• $\tan(-\theta) = -\tan \theta$ (tangent is odd).

Geometrically, negative angles go clockwise, giving a point reflected across the $x$-axis.

Periodicity

Rotating a full turn ($360^\circ$ or $2\pi$ radians) brings you back to the same point on the unit circle:

• $\sin(\theta + 2\pi) = \sin\theta$,
• $\cos(\theta + 2\pi) = \cos\theta$,
• $\tan(\theta + \pi) = \tan\theta$ (tangent repeats every $\pi$).

This is visible on the unit circle: the same angle plus any integer multiple of $2\pi$ lands on the same point.

Using the Unit Circle to Visualize Trigonometric Ratios

On the unit circle, for angle $\theta$:

• The $x$-coordinate of the point is the cosine.
• The $y$-coordinate is the sine.
• The slope of the radius (rise over run) is $\tan\theta$ whenever $\cos\theta \neq 0$.

This geometric picture allows you to:

• see why sine and cosine stay between $-1$ and $1$ (they are coordinates on a circle of radius 1),
• see when tangent is undefined (when the radius is vertical, i.e., $x = 0$).

Summary of Key Facts

• The unit circle is the circle $x^2 + y^2 = 1$ centered at the origin.
• For any angle $\theta$, the corresponding point on the unit circle is $(\cos\theta,\ \sin\theta)$.
• Quadrants determine the signs of $\sin\theta$, $\cos\theta$, and $\tan\theta$.
• Special angles ($0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$ and their related angles) have exact coordinate values using $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, and $\frac{\sqrt{3}}{2}$.
• Symmetry and reference angles let you deduce many values from a small set of basic ones.
• Sine and cosine repeat every $2\pi$; tangent repeats every $\pi$.

The unit circle picture ties together trigonometric ratios, angles, and coordinates, and it is the main visual tool for understanding trigonometric functions.