The Unit Circle & Radians

Radians: angle as arc length

Every circle is made of the same stuff: a center, a constant distance (radius), and a round boundary. If we agree to use a circle of radius $1$ — the unit circle — then angles and arc lengths are the same number.

Definition. One radian is the angle subtended at the center of a circle by an arc whose length equals the radius.

On the unit circle (radius $= 1$): an arc of length $s$ subtends an angle of exactly $s$ radians. The full circumference is $2\pi$, so a full turn is $2\pi$ radians. That's it. The whole number circle has $2\pi$ worth of angle in it, not $360$.

Why is this not arbitrary? Because the formula for arc length is: $s = r\theta \quad (\theta \text{ in radians}),$ with NO conversion factor. In degrees, you'd need $s = r\theta(\pi/180)$. That $\pi/180$ is historical baggage, a constant that exists solely to apologize for a bad choice. Radians make the formula $s = r\theta$ — clean, honest, no junk. This is what mathematics looks like when the units are chosen by the structure of the problem, not by some ancient dude counting fingers.

Converting between radians and degrees

The bridge: $2\pi \text{ rad} = 360°$, so $\pi \text{ rad} = 180°$.

$\text{degrees} \to \text{radians:} \quad \theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}.$ $\text{radians} \to \text{degrees:} \quad \theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180}{\pi}.$

The critical values, which you should know cold:

The unit circle: the real home of sin and cos

Here is the upgrade. Place a circle of radius $1$ centered at the origin. Pick any angle $\theta$, measured counterclockwise from the positive $x$-axis. Draw the radius to that angle. The endpoint of that radius is the point

$(\cos\theta,\ \sin\theta).$

This is the definition. Not a ratio of triangle sides — an actual coordinate on the unit circle.

Why is this consistent with the triangle definition? If $0 < \theta < 90°$, drop a perpendicular from the endpoint to the $x$-axis. You get a right triangle with hypotenuse $1$ (the radius), horizontal leg $x$ (adjacent to $\theta$), and vertical leg $y$ (opposite $\theta$). So:

$\cos\theta = \frac{\text{adj}}{\text{hyp}} = \frac{x}{1} = x, \qquad \sin\theta = \frac{\text{opp}}{\text{hyp}} = \frac{y}{1} = y.$

The triangle definition and the circle definition agree in the first quadrant — and the circle definition then extends to all angles naturally, including $90°$, $180°$, negative angles, and angles bigger than $360°$. The triangle was the larval form. This is the goddamn butterfly. Drag the widget and feel the upgrade:

Drag the angle around and watch the coordinates change:

Signs by quadrant

The unit circle tells you the sign of sin and cos everywhere:

• Quadrant I ($0 < \theta < \pi/2$): $x > 0$, $y > 0$. Both positive.
• Quadrant II ($\pi/2 < \theta < \pi$): $x < 0$, $y > 0$. Cos negative, sin positive.
• Quadrant III ($\pi < \theta < 3\pi/2$): $x < 0$, $y < 0$. Both negative.
• Quadrant IV ($3\pi/2 < \theta < 2\pi$): $x > 0$, $y < 0$. Cos positive, sin negative.

Mnemonic: ASTC ("All Students Take Calculus" — or in my lab, "All Sine-butchering students Take Consequences") — All, Sine, Tangent, Cosine are positive in quadrants I, II, III, IV respectively.

Reference angles

For any angle $\theta$, the reference angle is the acute angle it makes with the nearest part of the $x$-axis. If you know the reference angle $\alpha$, then $|\sin\theta| = \sin\alpha$ and $|\cos\theta| = \cos\alpha$; just pick the sign based on the quadrant.

Example: $\theta = 150°$. Reference angle: $180° - 150° = 30°$. Quadrant II, so $\cos < 0$ and $\sin > 0$: $\cos 150° = -\cos 30° = -\frac{\sqrt{3}}{2}, \qquad \sin 150° = \sin 30° = \frac{1}{2}.$

Negative angles and angles beyond 360°

On the unit circle:

• Negative angles go clockwise. $\theta = -\pi/4$ is the same point as $\theta = 7\pi/4$.
• Angles beyond $2\pi$ just wrap around. $\sin(\theta + 2\pi) = \sin\theta$ for all $\theta$ — periodicity.

Sin and cos as waves

Unroll the unit circle onto a horizontal axis: let $\theta$ run from $0$ to $2\pi$ (one full revolution). Plot the $y$-coordinate (sin) against $\theta$: you get a sine wave — it rises from $0$ to $1$ at $\pi/2$, falls back to $0$ at $\pi$, dips to $-1$ at $3\pi/2$, and returns to $0$ at $2\pi$. The cosine wave is the same shape, shifted left by $\pi/2$.

Both waves have period $2\pi$, amplitude $1$, and repeat forever in both directions. The full story of periodic functions is its own lesson; the point here is that the circle IS the wave, just coiled up.