The Unit Circle
One circle of radius 1, and every value of sine, cosine and tangent you are ever asked to remember. Drag the angle and watch where the numbers come from.
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Almost everything people find hard about trigonometry is really a memory problem: too many values, too many signs, no way to check whether the one you have just written down is plausible. The unit circle is the fix. It is a single picture that holds every value you need and makes a wrong sign look obviously wrong.
Drag the point below. Watch the two dashed legs as you go round.
Drag the point. The green leg is cos, the red leg is sin - which is why the point is (cos, sin).
Why the point is (cos, sin)
The circle has radius 1, and that is the whole trick.
Draw a line from the centre out to the circle at some angle, then drop a vertical line from that point down to the x-axis. You now have a right triangle whose hypotenuse is the radius, so the hypotenuse is 1.
In that triangle, cosine is adjacent over hypotenuse. The adjacent side is the horizontal leg, and the hypotenuse is 1, so cosine is the horizontal leg — dividing by 1 changes nothing. The same argument makes sine the vertical leg.
So the coordinates of the point are exactly (cos(x), sin(x)) — the horizontal distance first, then the vertical one. That is why the green leg in the figure is labelled cos and the red one sin, and it is why the order is cos first: x before y, horizontal before vertical.
Tangent is the third one, and it is not a length on the picture — it is the ratio sin(x)/cos(x), which is the slope of the line you drew.
The three numbers you actually need
Set the figure to 30, 45 and 60 degrees in turn and look at the values:
| angle | cos | sin |
|---|---|---|
| 30° | sqrt(3)/2 | 1/2 |
| 45° | sqrt(2)/2 | sqrt(2)/2 |
| 60° | 1/2 | sqrt(3)/2 |
Three distinct values — 1/2, sqrt(2)/2 and sqrt(3)/2 — and notice that 30° and 60° are the same pair swapped. That is not a coincidence: the two angles add to 90°, so their triangles are the same triangle looked at from the other end.
Everything in the other three quadrants is one of those three numbers with a sign in front. Which sign is decided by where you are, and you can read it straight off the figure: cos is the horizontal distance, so it is negative on the left half of the circle. Sin is the vertical distance, so it is negative on the bottom half. Nothing to memorise — just look at which way the leg points.
Degrees and radians are the same angle
Switch the figure to radians and drag round again. Nothing about the picture changes; only the label does.
A full turn is 360°, and it is also 2pi radians — because the circumference of a circle of radius 1 is 2pi, so "one radian" means "an arc one radius long". Half a turn is 180° or pi. From there the special angles are obvious fractions of a half turn: 30° is a sixth of 180°, so it is pi/6; 45° is a quarter, pi/4; 60° a third, pi/3.
Radians look harder and are actually easier, because the denominator tells you how many of that angle fit into a half turn.
Where tangent breaks
Drag to 90° and the tan readout says undefined. That is not a limitation of the widget — it is arithmetic. At 90° the point is straight up at (0, 1), so cos(x) is 0 — and tan is sin(x)/cos(x). There is no number equal to 1 divided by 0.
The same happens at 270°. Everywhere else tan is a perfectly ordinary number, and at 45° it is exactly 1 — which makes sense, since sin and cos are equal there and anything divided by itself is 1.
Try one
Set the figure aside for a moment and answer from memory. If you get it wrong, the circle is right there.
If you had to look, that is worth knowing — it means the picture has not become automatic yet, and drilling a dozen of these is what fixes it.
Common questions
- What is the unit circle?
- It is the circle of radius 1 centred on the origin. Its usefulness comes from one fact: if you draw a line from the centre at an angle and mark where it crosses the circle, that point's coordinates are exactly (cos of the angle, sin of the angle). So the circle turns a question about triangles into a question about coordinates, which is much easier to picture.
- Why is the point on the circle (cos, sin) and not (sin, cos)?
- Because cosine measures horizontal distance and the x-coordinate is the horizontal one. Drop a vertical line from the point to the x-axis and you get a right triangle whose hypotenuse is 1. The horizontal leg is adjacent to the angle, so it is cos over 1, which is just cos. The vertical leg is opposite, so it is sin. The x-coordinate is therefore cos and the y-coordinate is sin, in that order.
- Do I have to memorise the whole unit circle?
- No, and trying to is the usual mistake. There are only three values you need - 1/2, √2/2 and √3/2 - and one rule for signs. Everything else is those three numbers with a plus or minus in front, decided by which quadrant you are in. Learn the first quadrant and the sign rule and you have all sixteen angles.
- How do I convert between degrees and radians?
- A full turn is 360 degrees and also 2π radians, so half a turn is 180 degrees and π radians. That gives the conversion: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. In practice you rarely calculate it - 30, 45, 60 and 90 degrees are π/6, π/4, π/3 and π/2, and every other special angle is a multiple of those.
- Why is tan undefined at 90 degrees?
- Because tan is sin divided by cos, and at 90 degrees cos is 0. Dividing by zero is not a number, so tan has no value there. On a graph of tan you see this as a vertical asymptote: as the angle approaches 90 degrees the value shoots off without ever arriving anywhere.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
