Sine and Cosine on the Unit Circle

Part 12, Chapter 12: Trigonometry of the Unit Circle

Learning objectives

Define sine and cosine using the unit circle
State and apply the Pythagorean identity
Evaluate sin and cos at standard angles
Determine the sign of sin and cos in each quadrant

Right-triangle trigonometry runs out of room. "Opposite over hypotenuse" makes perfect sense when the angle is between $0$ and $\pi/2$ , but what is $\sin(120°)$ ? There is no right triangle with a $120°$ interior angle. To talk about any angle, obtuse, reflex, negative, beyond a full turn, we need a definition that does not depend on triangles at all. The unit circle gives us one. Sine and cosine become the two coordinates of a point sweeping around a circle of radius $1$ , and every old triangle fact survives as a special case.

The definition

Draw the unit circle: $x^2 + y^2 = 1$ , centred at the origin. Start at the point $(1, 0)$ on the positive $x$ -axis. Sweep counterclockwise through angle $\theta$ radians. You land on a point $P_\theta$ theta on the circle. The cosine of $\theta$ is the $x$ -coordinate of $P_\theta$ theta, and the sine is the $y$ -coordinate:

$P_\theta = (\cos\theta,\; \sin\theta)$ theta=(costheta,;sintheta)

That is the whole definition. It is purely geometric and assumes nothing about triangles. For $0 < \theta < \pi/2$ it recovers the right-triangle ratios, because the radius, the $x$ -drop, and the $y$ -drop form a right triangle with hypotenuse $1$ .

Where this shows up

Signal Processing: Every audio signal can be decomposed into a sum of sines and cosines (Fourier transform); MP3 compression literally throws away the sine/cosine components your ear cannot hear.
Game Physics: Projectile trajectories use $\sin(\theta)$ and $\cos(\theta)$ to split initial velocity into horizontal and vertical components, every Angry Birds shot is two trig calls.
Astronomy: The position of any orbiting body at time $t$ involves $\sin$ and $\cos$ of the orbital angle $\omega t$ ; ephemerides for satellite tracking are tables of these values.

(Drag the orange point. Watch $\cos\theta$ trace out as the horizontal distance from the $y$ -axis and $\sin\theta$ as the vertical distance from the $x$ -axis. The readout shows both signed coordinates and the radian/degree readings.)

The Pythagorean identity

The point $(\cos\theta, \sin\theta)$ lives on the circle $x^2 + y^2 = 1$ , so substituting gives the most-used identity in trigonometry:

$\sin^2\theta + \cos^2\theta = 1 \quad \text{for every } \theta$

It is just the Pythagorean theorem applied to the radius. Memorise it; you will reach for it constantly.

The standard values

The exact values at $\theta = 0, \pi/6, \pi/4, \pi/3, \pi/2$ come from the geometry of the $30°-60°-90°$ and $45°-45°-90°$ triangles, scaled to fit inside the unit circle:

$\sin 0 = 0,\; \sin\tfrac{\pi}{6} = \tfrac{1}{2},\; \sin\tfrac{\pi}{4} = \tfrac{\sqrt{2}}{2},\; \sin\tfrac{\pi}{3} = \tfrac{\sqrt{3}}{2},\; \sin\tfrac{\pi}{2} = 1$

$\cos 0 = 1,\; \cos\tfrac{\pi}{6} = \tfrac{\sqrt{3}}{2},\; \cos\tfrac{\pi}{4} = \tfrac{\sqrt{2}}{2},\; \cos\tfrac{\pi}{3} = \tfrac{1}{2},\; \cos\tfrac{\pi}{2} = 0$

Notice the symmetry: as one rises, the other falls, which is the Pythagorean identity in disguise.

Signs by quadrant

The four quadrants split the unit circle into regions where each coordinate has a fixed sign:

Quadrant I ( $0 < \theta < \pi/2$ ): both positive.
Quadrant II ( $\pi/2 < \theta < \pi$ ): $\sin > 0$ , $\cos < 0$ .
Quadrant III ( $\pi < \theta < 3\pi/2$ ): both negative.
Quadrant IV ( $3\pi/2 < \theta < 2\pi$ ): $\sin < 0$ , $\cos > 0$ .

The mnemonic All Students Take Calculus records the positive-function pattern: All positive in QI, Sine positive in QII, Tangent positive in QIII, Cosine positive in QIV.

Try it

Set $\theta =

\\sin\\theta

^\circ $. Predict first: what is$ \sin 30^\circ $? Rotate to$ 30^\circ = \pi/6 $and verify the readout shows$ \sin\theta = 0.5$ exactly.

Rotate to $\theta =$^\circ $and verify$ \sin^2\theta + \cos^2\theta = 1 $using the exact values$ \sin\theta = \cos\theta = \sqrt{2}/2$.

Pause: which is the $x$ -coordinate, sine or cosine? If you have to think about it longer than two seconds, this is the section to overlearn. Cosine is $x$ . Sine is $y$ . The alphabetic order matches the coordinate order.

Try it in code

A trap to watch for

Beginners routinely write $\sin\theta$ for the $x$ -coordinate and $\cos\theta$ for the $y$ -coordinate. This is a coin-flip mistake that scrambles every calculation downstream. The fix is to anchor the names to the unit circle picture: cosine catches the shadow on the x-axis when light shines straight down; sine catches the shadow on the y-axis when light shines straight across. Or, even simpler, memorise: (cos, sin) in alphabetical order matches (x, y) in alphabetical order. Until the association is automatic, refer back to the widget above.

What you now know

You can produce $\sin\theta$ and $\cos\theta$ for any angle by reading coordinates off the unit circle, apply the Pythagorean identity to solve for one given the other and the quadrant, and recite the standard-angle values. The next section drops the static picture and watches what happens when $\theta$ varies, producing the wave-shaped graphs of $y = \sin x$ and $y = \cos x$ .

Quick check

Mark section complete →

References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 11, §2, sine and cosine defined coordinate-wise on the unit circle.
Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Chapter 15 derives the analytic properties of sin and cos from the unit-circle definition.
Apostol, T. M. (1967). Calculus, Volume 1 (2nd ed.). Wiley. §2.5 develops trigonometric functions via arc length.