An angle is generated by rotating a ray (the initial side) about its endpoint (the vertex) until it reaches a final position (the terminal side). The direction of rotation gives the angle its sign: anticlockwise rotation is positive, clockwise rotation is negative. One full sweep of the ray back to its own starting position is one complete rotation.
Three systems for measuring an angle:
- Sexagesimal (degree) system — a right angle split into 90 degrees (∘); 1∘=60 minutes (′); 1′=60 seconds (′′).
- Centesimal system — a right angle split into 100 grades (g); each grade into 100 minutes; each of those into 100 seconds. Rarely used in practice.
- Circular (radian) system — measured via an arc-length-to-radius ratio, developed fully below. The system used throughout higher mathematics.
Degree measure in detail. One complete rotation is 360∘, so 1∘=3601 of a full turn; 1′=601 of a degree; 1′′=601 of a minute =36001 of a degree. Two angles are congruent if equal in measure, complementary if they sum to 90∘, supplementary if they sum to 180∘, and conjugate if both lie in [0∘,360∘) and sum to 360∘.
Standard position and quadrants. An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis; it is then a first/second/third/fourth-quadrant angle according to where its terminal side falls. An angle whose terminal side falls exactly on an axis is a quadrantal angle — always a multiple of 90∘ (0∘,90∘,180∘,270∘,360∘,…).
Coterminal angles. Two angles in standard position with the same terminal side (however many extra full rotations were used to get there) are coterminal. If α,β are coterminal, β=α+k(360∘) for some integer k. To find the coterminal angle of any given angle inside the standard range [0∘,360∘), repeatedly add or subtract 360∘ until the result lands in that range.
Why degrees aren't enough. Right-triangle trigonometry, and degree measure along with it, only ever produces acute angles. But calculus, physics and chemistry need trigonometric functions defined on the whole real line. The fix is to measure an angle using a length ratio instead of an arbitrary human convention like "360 parts to a turn."
Definition — the radian. For a circle of radius r, let an angle θ at the centre subtend an arc of length s. Then the radian measure of θ is
θ=rs radians,i.e.s=rθ.
Because all circles are similar, this ratio depends only on the angle, never on which particular circle is used to measure it. When s=r exactly, θ=1 radian — the angle subtended by an arc exactly as long as the radius. Since s and r are both lengths, θ is a pure (unitless) number, which is why a plain number like 3π or 2, with no symbol attached, is automatically read as being in radians.
Degree ↔ radian conversion. A full rotation is both 360∘ (by definition of the degree) and 2π radians (the circumference of the unit circle, since s=rθ with r=1,θ=2π gives s=2π). Equating the two:
2π rad=360∘ ⟹ π rad=180∘ ⟹ 1∘=180π rad,1 rad=(π180)∘.
More generally, x∘=180πx radians and x radians =(π180x)∘.
Why radians are preferred. Formulas involving angles are simply cleaner in radian measure — compare the sector-area formula 360∘πr2θ (degrees) with 21r2θ (radians): no leftover π or 360 in the radian version. This is exactly why calculus formulas (derivatives/integrals of sin,cos, etc.) are only clean when the angle is in radians.
Worked conversion example (own numbers). Convert 150∘ to radians and 45π radians to degrees.
- 150∘=180π×150=65π radians.
- 45π radians =π180×45π=4180×5=225∘.
A quick way to remember which way a conversion factor goes: 180∘=π≈3.14, a smaller number, so converting degrees → radians means multiplying by the small factor π/180, and radians → degrees means multiplying by the large factor 180/π.