1. The unit circle
The circle of radius $1$ centered at the origin in the $xy$-plane — the set of all points $(x, y)$ satisfying $x^2 + y^2 = 1$.
Pick any point $P$ on this circle. Draw the segment from the origin to $P$, and measure the angle $\theta$ that segment makes with the positive $x$-axis (counter-clockwise is positive, the standard convention). Then the coordinates of $P$ are, by definition:
$$ P = (\cos\theta, \;\sin\theta) $$Read that again. The $x$-coordinate is $\cos\theta$. The $y$-coordinate is $\sin\theta$. The unit circle is the picture that turns those two functions from "ratios in a right triangle" into "coordinates on a curve" — and once you internalize this, almost every identity in trig becomes a geometric observation.
Tangent rides along for the ratio:
$$ \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x} $$which is the slope of the line from the origin through $P$.
2. Why this generalizes trig beyond right triangles
In a right triangle, $\sin\theta$ is "opposite over hypotenuse" and $\cos\theta$ is "adjacent over hypotenuse." That works beautifully for $\theta$ between $0°$ and $90°$ — but then it stops working. What's the "opposite side" of a $135°$ angle? Of $-30°$? Of $720°$? The right triangle vanishes.
The unit-circle definition doesn't care. Rotate the radius to any angle you like — positive, negative, larger than a full turn, irrational, doesn't matter. The radius lands somewhere. That landing point has an $x$-coordinate and a $y$-coordinate. We declare those to be $\cos\theta$ and $\sin\theta$. End of definition.
For $0 < \theta < 90°$ the unit-circle definition agrees exactly with the right-triangle one — drop a perpendicular from $P$ to the $x$-axis and you get a right triangle with hypotenuse $1$, opposite side $\sin\theta$, adjacent side $\cos\theta$. Outside that range, the unit circle keeps going where the triangle gave up.
This is also why we move from degrees to radians. A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. On the unit circle that means $\theta$ in radians equals the arc length traveled from $(1, 0)$ around to $P$. A full trip around the circle is $2\pi$ radians, and the angle and the distance traveled become the same number.
The picture worth a thousand words
Every dot on that circle is a tiny lookup table: angle in, $(\cos, \sin)$ out. The whole of trigonometric values at the "special" angles lives in this one diagram.
3. Signs of trig functions by quadrant
The plane is sliced by the axes into four quadrants. Because $\cos\theta$ is the $x$-coordinate and $\sin\theta$ is the $y$-coordinate of the point on the circle, the signs of the trig functions are nothing more than the signs of $x$ and $y$ in each quadrant.
| Quadrant | $\theta$ | $x = \cos\theta$ | $y = \sin\theta$ | $\tan\theta = y/x$ |
|---|---|---|---|---|
| Q1 | $0$ to $\tfrac{\pi}{2}$ | + | + | + |
| Q2 | $\tfrac{\pi}{2}$ to $\pi$ | − | + | − |
| Q3 | $\pi$ to $\tfrac{3\pi}{2}$ | − | − | + |
| Q4 | $\tfrac{3\pi}{2}$ to $2\pi$ | + | − | − |
Read down the columns: $\sin$ is positive in Q1 & Q2 (top half of the plane, where $y > 0$). $\cos$ is positive in Q1 & Q4 (right half, where $x > 0$). $\tan$ is positive when $\sin$ and $\cos$ agree in sign — Q1 (both +) and Q3 (both −).
"All Students Take Calculus." Walk counter-clockwise from Q1: in Q1 all functions are positive; in Q2 only sine is positive; in Q3 only tangent; in Q4 only cosine. Picture the four letters written one in each quadrant and you have a permanent cheat sheet.
The ASTC diagram
4. Reference angles
If you know the trig values at the special angles in the first quadrant, you know them everywhere. The bridge is the reference angle.
The acute angle between the terminal side of $\theta$ and the $x$-axis. Always between $0$ and $\tfrac{\pi}{2}$ (or $0°$ and $90°$). Call it $\theta_\text{ref}$.
The magnitudes of $\sin$, $\cos$, and $\tan$ at $\theta$ are exactly their values at $\theta_\text{ref}$. Only the sign changes, and the sign is determined by the quadrant. The recipe:
- Find which quadrant $\theta$ lives in.
- Find $\theta_\text{ref}$ — the acute angle to the nearest part of the $x$-axis.
- Look up the magnitude using $\theta_\text{ref}$.
- Attach the correct sign using ASTC.
The reference angle for $\theta$ in each quadrant:
| Quadrant of $\theta$ | $\theta_\text{ref}$ (in radians) | $\theta_\text{ref}$ (in degrees) |
|---|---|---|
| Q1 | $\theta$ | $\theta$ |
| Q2 | $\pi - \theta$ | $180° - \theta$ |
| Q3 | $\theta - \pi$ | $\theta - 180°$ |
| Q4 | $2\pi - \theta$ | $360° - \theta$ |
Example. To find $\sin(210°)$: that's in Q3 (between $180°$ and $270°$). The reference angle is $210° - 180° = 30°$. So $|\sin(210°)| = \sin(30°) = \tfrac{1}{2}$. ASTC says sine is negative in Q3. Therefore $\sin(210°) = -\tfrac{1}{2}$.
5. Periodicity
The unit circle makes this almost too obvious. If you rotate by a full turn — $2\pi$ radians, or $360°$ — you end up at exactly the same point you started. So the coordinates didn't change, and therefore:
$$ \sin(\theta + 2\pi) = \sin\theta, \qquad \cos(\theta + 2\pi) = \cos\theta $$That's what we mean when we say sine and cosine are periodic with period $2\pi$. Going around the circle is the literal act of repeating.
Tangent is different. Recall $\tan\theta = \sin\theta / \cos\theta$. Rotating by just $\pi$ — a half turn — flips the signs of both $\sin$ and $\cos$:
$$ \sin(\theta + \pi) = -\sin\theta, \qquad \cos(\theta + \pi) = -\cos\theta $$The two minus signs cancel in the ratio:
$$ \tan(\theta + \pi) = \frac{-\sin\theta}{-\cos\theta} = \tan\theta $$So tangent comes back to itself after only half a turn. Its period is $\pi$, not $2\pi$.
It's tempting to assume "all trig functions have period $2\pi$." They don't. $\sin$ and $\cos$ do; $\tan$ and $\cot$ have period $\pi$. This catches people out in calculus, in Fourier series, and especially when solving equations like $\tan x = 1$ (which has solutions every $\pi$, not every $2\pi$).
6. The famous table of exact values
You will look these up endlessly until you don't. They're the values you can write down without a calculator, because they fall on the special points of the unit circle — the corners of a $30$-$60$-$90$ or $45$-$45$-$90$ triangle inscribed in it.
| Degrees | Radians | $\cos\theta$ | $\sin\theta$ | $\tan\theta$ |
|---|---|---|---|---|
| $0°$ | $0$ | $1$ | $0$ | $0$ |
| $30°$ | $\tfrac{\pi}{6}$ | $\tfrac{\sqrt{3}}{2}$ | $\tfrac{1}{2}$ | $\tfrac{\sqrt{3}}{3}$ |
| $45°$ | $\tfrac{\pi}{4}$ | $\tfrac{\sqrt{2}}{2}$ | $\tfrac{\sqrt{2}}{2}$ | $1$ |
| $60°$ | $\tfrac{\pi}{3}$ | $\tfrac{1}{2}$ | $\tfrac{\sqrt{3}}{2}$ | $\sqrt{3}$ |
| $90°$ | $\tfrac{\pi}{2}$ | $0$ | $1$ | undefined |
| $180°$ | $\pi$ | $-1$ | $0$ | $0$ |
| $270°$ | $\tfrac{3\pi}{2}$ | $0$ | $-1$ | undefined |
| $360°$ | $2\pi$ | $1$ | $0$ | $0$ |
Look at the $\sin$ column for $0°, 30°, 45°, 60°, 90°$: $\;\tfrac{\sqrt{0}}{2}, \tfrac{\sqrt{1}}{2}, \tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{3}}{2}, \tfrac{\sqrt{4}}{2}$. The numerators are just $\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}$. The $\cos$ column is the same list reversed. That's the entire first-quadrant table, recoverable from a single pattern.
Combine the table with reference angles and ASTC, and you can compute $\sin$, $\cos$, $\tan$ of any "nice" multiple of $30°$ or $45°$ in your head. The unit circle isn't a chart you memorize — it's a small set of moves you internalize.