Radians & the Unit Circle
Measure angles by the arc they cut, and trigonometry escapes the right triangle.
Take the radius of a circle and lay it along the rim.
Bend it round the edge, mark where it ends, and start again from there.
Six of them fit, and there is a short piece left over — about a quarter of another radius.
That is true for every circle ever drawn. Big or small, it is always six and a bit.
The "six and a bit" is 2π ≈ 6.28. One radian is the angle whose arc equals the radius — an angle measured by the circle itself, not by a number somebody chose.
Converting between degrees and radians
A half turn is 180° and also π radians, so π radians = 180° is the only fact you need.
To go from degrees to radians, multiply by π180. To come back, multiply by 180π.
So 60° = 60 · π180 = π3, and 3π4 = 135°.
Put pi where you want it
Multiplying by π180 brings π in; by 180π takes it out. Ask which one your answer should have.
Coordinates are cosine and sine
On the unit circle, the point at angle θ has coordinates (cosθ, sinθ). That is the definition, not a coincidence.
It agrees with the triangle version: the hypotenuse is 1, so adjacent over hypotenuse is just the adjacent side.
And it keeps working past 90°, where no right triangle exists — which is the whole point of moving to the circle.
x is cos, y is sin
Alphabetical order in both: x before y, cos before sin. That is enough to remember it for good.
Signs by quadrant
The signs of cos and sin just follow the signs of x and y in each quadrant.
Quadrant II has x negative and y positive, so cosine is negative and sine is positive there.
Nothing needs memorising beyond the coordinate plane you have used for years.
Sketch, do not recite
Draw the angle, see which quadrant it lands in, read off the signs of x and y. A picture beats a mnemonic you might misremember.
Reference angles
Every angle has a reference angle: the acute angle it makes with the x-axis.
150° has reference angle 30°, so sin 150° = sin 30° = 12, with the sign fixed by the quadrant.
That is why knowing 30°, 45° and 60° is enough to handle every special angle on the circle.
Size from the reference, sign from the quadrant
Two separate questions. Answer the size first, then attach the sign — mixing them is where errors come from.
Arc length and coterminal angles
Arc length is beautifully simple in radians: s = rθ, with no conversion factor at all.
That simplicity is the reason radians exist. In degrees the same formula needs π180 bolted on.
Adding 2π to an angle lands you in the same place, so π6 and 13π6 have identical trig values.
The formula demands radians
Using s = rθ with θ in degrees gives an answer about 57 times too large. Convert first, always.
Worked examples
Degrees to radians
Convert 135° to radians.
- Multiply by π180: 135 · π180. This is the factor that brings π in.
- Simplify 135180 = 34. Both divide by 45.
- So 135° = 3π4. Sensible: 135° is three quarters of 180°.
3π4
A value past 90 degrees
Find cos 210° exactly.
- 210° is in quadrant III, 30° past 180°. The reference angle is 30°.
- cos 30° = √32. A special angle worth knowing.
- In quadrant III, x is negative, so cos 210° = -√32. Size from the reference, sign from the quadrant.
-√32
Arc length
A circle has radius 9 cm. Find the arc cut by an angle of 2π3.
- The angle is already in radians, so s = rθ applies directly. No conversion needed.
- s = 9 · 2π3 = 6π cm. The 9 and the 3 cancel.
- About 18.8 cm. A third of the way round a circle of circumference 18π — consistent.
6π cm
Words to know
- Radian
- The angle whose arc length equals the radius. A full turn is 2π radians.
- Unit circle
- The circle of radius 1 centred at the origin.
- Reference angle
- The acute angle between the terminal side and the x-axis.
- Coterminal angles
- Angles that finish in the same place, differing by whole turns.
- Terminal side
- Where the rotating arm ends up after turning through the angle.
Mistakes to watch for
Converting the wrong way. To turn 60° into radians multiply by π180, giving π3. Using 180π instead gives about 3438, which is not an angle anybody meant.
Using s = rθ with degrees. The formula is built for radians, so an angle of 60 gives an arc 57 times too long. Convert before substituting.
Getting the sign right but the size wrong, or the reverse. sin 150° is +12: the reference angle 30° gives the size, and quadrant II gives the plus. Both questions must be answered separately.
Swapping cosine and sine on the circle. The point is (cosθ, sinθ) — x first. Reading it the other way reflects every answer across the line y = x.
Thinking a radian is a kind of degree. It is a pure ratio of arc to radius, which is why π appears without any units attached.
The 360 in a circle is Babylonian, picked because 360 divides neatly and roughly matches the days in a year. There is nothing mathematical about it.
Radians make calculus work. The derivative of sin x is cos x only in radians; in degrees an ugly constant appears and never leaves.
Your calculator has both modes because both are in daily use — surveyors in degrees, physicists in radians.