JOVANA
Explore Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

The Unit Circle Sets Trig Free

SOHCAHTOA only speaks the language of acute angles inside a right triangle — but the world has angles of 0, 120, even 400 degrees. The unit circle keeps the same sine and cosine while quietly dropping the triangle, so trig finally works for every angle there is.

Where SOHCAHTOA runs out of road

From the previous guide you can already say SOHCAHTOA in your sleep: in a right triangle, sin theta is opposite over hypotenuse, cos theta is adjacent over hypotenuse, tan theta is opposite over adjacent. It is a beautiful tool — but look hard at it and you notice a cage. Every one of those ratios lives inside a right triangle, and a right triangle's other two angles must be acute, strictly between 0 and 90 degrees. So as defined, sin 90 degrees is awkward (the triangle has collapsed) and sin 120 degrees is simply meaningless. There is no right triangle with a 120-degree angle in it.

Yet angles bigger than 90 degrees are everywhere the moment you leave the textbook triangle. A wheel turning past the top of its spin, a pendulum swinging back the other way, a boat on a bearing of 200 degrees, the alternating current in a wall socket — all of them want a sine and a cosine of angles that no acute triangle can hold. The honest situation is this: SOHCAHTOA is not wrong, it is just narrow. We need a definition that agrees with it perfectly on acute angles but keeps making sense when the angle grows past 90, past 180, past 360, or even goes negative.

One circle, radius one, centred at the origin

The unit circle is the simplest circle you could name: centre at the origin of the coordinate plane, radius exactly 1. Now the key move. Start a ray along the positive x-axis and sweep it counterclockwise through an angle theta. Where the ray crosses the circle, it touches a single point P. The definition that sets everything free is this: cos theta is the x-coordinate of P, and sin theta is the y-coordinate of P. That is the whole idea. The point P, riding on the circle, is the ordered pair (cos theta, sin theta).

Check that this agrees with the old triangle for an acute theta. Drop a perpendicular from P straight down to the x-axis, hitting it at a point Q. Triangle OPQ is a right triangle: the hypotenuse OP has length 1 (it is a radius), the horizontal leg OQ has length equal to P's x-coordinate, and the vertical leg PQ has length equal to P's y-coordinate. By SOHCAHTOA, cos theta = adjacent/hypotenuse = OQ/1 = OQ = x, and sin theta = opposite/hypotenuse = PQ/1 = PQ = y. Identical. The unit circle did not change a single value for an acute angle — it just stopped needing the triangle to be there.

Because the hypotenuse is locked at 1, the messy denominators vanish and the coordinates are the trig values directly. That is the quiet genius of choosing radius 1 rather than any old circle: it turns sine and cosine from ratios into plain positions. Want tan theta? It is still opposite over adjacent, which here is just sin theta / cos theta = y/x — the slope of the ray OP. Every value you knew is still here; it has simply been given coordinates to stand on.

Now the angle can grow past 90 — and go negative

Here is the freedom we paid for. The point P keeps gliding around the circle no matter how far theta sweeps, so its coordinates — and therefore sin theta and cos theta — are defined for every angle. At theta = 90 degrees, P sits at the top, (0, 1), so cos 90 = 0 and sin 90 = 1, no collapsed triangle required. At theta = 180 degrees, P is at (-1, 0): cos 180 = -1, sin 180 = 0. At 270 degrees, P is at (0, -1). The values march around the circle as smoothly as the point does.

Crucially, the signs now take care of themselves. In the second quadrant (angles between 90 and 180), P has a negative x and a positive y, so cosine is negative while sine stays positive — that is why cos 120 degrees is negative, not because of any rule to memorise but because the point genuinely sits to the left of the y-axis. Sweep clockwise instead and you get negative angles: theta = -30 degrees lands P below the axis, giving sin(-30) = -1/2. The single picture explains all the sign patterns that look like arbitrary trivia in a table.

The Pythagorean identity, hiding in plain sight

Because P = (cos theta, sin theta) lives on a circle of radius 1, its coordinates must satisfy the circle's own equation, x^2 + y^2 = 1. Substitute and you get the single most-used fact in all of trigonometry, the Pythagorean identity: cos^2 theta + sin^2 theta = 1, true for every angle theta whatsoever. It is not a new mystery — it is the Pythagorean theorem applied to the right triangle OPQ, whose legs are cos theta and sin theta and whose hypotenuse is the radius 1.

P = (cos theta, sin theta)  on the unit circle x^2 + y^2 = 1

   y |       . P = (cos t, sin t)
     |      /|
   1 +    / |  sin t   (vertical leg)
     |   /  |
     |  / t |
  ---+--------+----  x
     O   cos t  Q
     |  (horizontal leg)

x^2 + y^2 = 1   =>   (cos theta)^2 + (sin theta)^2 = 1
The unit circle with the right triangle OPQ inside it. Reading P's coordinates off the axes gives cosine and sine; the circle equation x^2 + y^2 = 1 instantly becomes the Pythagorean identity.

Notice how much this one relation does. Tell me sin theta and the quadrant, and the identity hands you cos theta up to sign (then the quadrant fixes the sign). It is the reason the trig functions are knotted together rather than independent — knowing one, plus a little location information, pins the others. We write cos^2 theta to mean (cos theta)^2; the square sits on the value, not on the angle. This identity, born from a circle of radius 1, will be the seed of nearly every further identity you meet in the last guide of this rung.

Building the circle's landmark points

You do not memorise the unit circle; you build it from a handful of triangles you already trust. The whole circle is pinned down by the special right triangles from the earlier rung — the 30-60-90 and the 45-45-90. Their side ratios become the coordinates of P at the famous angles, and the four-fold symmetry of the circle copies each first-quadrant point into the other three quadrants, flipping signs as it goes. Master one quadrant honestly and the rest are reflections.

  1. Anchor the four axis points first, where no triangle is needed: theta = 0 gives P = (1, 0); 90 degrees gives (0, 1); 180 gives (-1, 0); 270 gives (0, -1).
  2. For 45 degrees, use the 45-45-90 triangle: both legs equal, so x = y, and with x^2 + y^2 = 1 each is (sqrt 2)/2. So P = ((sqrt 2)/2, (sqrt 2)/2).
  3. For 30 degrees, the 30-60-90 triangle gives the short leg (here the height) = 1/2 and the long leg = (sqrt 3)/2, so P = ((sqrt 3)/2, 1/2). For 60 degrees, swap the coordinates: P = (1/2, (sqrt 3)/2).
  4. Reflect into the other quadrants using the reference angle: 150 degrees mirrors 30 across the y-axis to (-(sqrt 3)/2, 1/2); 210 degrees lands in quadrant III at (-(sqrt 3)/2, -1/2). Same numbers, signs set by quadrant.

One honest caveat before you go: we have measured angles in degrees throughout, which is comfortable but not the measure the circle itself prefers. The unit circle has circumference 2 pi, and that number is begging to become the unit of angle — a full turn as 2 pi rather than 360. That switch, to radians, is exactly what the sine, cosine and tangent functions secretly run on, and it is the subject of the very next guide. The circle you just built is the stage; radians are about to give it its natural ruler.