Let the ray spin — any number of turns, either direction — and measure the turn by arc.
Back in 25.1 an angle had to be one of the two acute corners of a right triangle, trapped between 0° and 90°. That cage is too small. A clock's hand sweeps past 90°, past 180°, past a full turn and around again; a wheel turns thousands of times. To let trigonometry describe spinning, vibrating, repeating things, we redefine an angle as a rotation: park a ray on the positive x-axis, then turn it. Turn it counter-clockwise and the angle is positive; turn it clockwise and it is negative; turn it as far as you like — 400°, −30°, three whole turns. Then we will measure that turn in a far more natural unit than the degree: the radian, where the length of arc the ray sweeps is the angle.
Draw a ray from the origin pointing east, along the positive x-axis. Call this the initial side. Now rotate it about the origin. Wherever it ends up is the terminal side, and the angle is the amount of turning that carried one onto the other.
Two choices come with every rotation. First, direction: a counter-clockwise turn is counted positive, a clockwise turn negative. (This matches the way we numbered the quadrants and the way angles open in the unit circle.) Second, amount: nothing stops the ray at 90° or even at 360°. Spin it a full turn and you are back where you started, having swept 360°; spin one-and-a-tenth turns and you have swept 396°. So an angle of rotation can be any real number of degrees — small, large, or negative.
An angle in standard position has its vertex at the origin and its initial side on the positive x-axis. The angle is the signed amount of rotation (CCW positive, CW negative) carrying the initial side to the terminal side — any real number of degrees.
Once the spinning stops, where the terminal side rests tells you the quadrant. If it lands strictly inside the upper-right region the angle is in Quadrant Ⅰ (between 0° and 90°), upper-left is Ⅱ (90°–180°), lower-left is Ⅲ (180°–270°), lower-right is Ⅳ (270°–360°). Land exactly on an axis — 0°, 90°, 180°, 270° — and the angle is quadrantal, belonging to no quadrant. Our helper GX.quadrant returns 1–4, or 0 on an axis.
Here is the beautiful part. A whole turn — 360° — brings the ray right back onto the same terminal side. So adding or subtracting any number of full turns changes the rotation count but not the final ray. Angles that share a terminal side are called coterminal. For instance
30°, 30° + 360° = 390°, 30° − 360° = −330° all point the same way.
To find the standard coterminal angle in [0°, 360°), add or subtract 360° until you land in that window — exactly what GX.coterminal does. Because coterminal angles share a terminal side, they share every trig value too: sin 390° = sin 30°. That single fact is what makes the trig functions periodic, the theme of 25.4.
Where does 760° rest? Subtract full turns: 760° − 360° = 400°, still too big; 400° − 360° = 40°. So 760° is coterminal with 40°, which sits in Quadrant Ⅰ. And −100°? Add 360°: −100° + 360° = 260°, which is in Quadrant Ⅲ.
The degree is an accident of history — somebody once chopped a full turn into 360 equal slices. There is a more honest unit, one the circle itself suggests. Stand at the centre of a circle of radius r and open an angle. The angle cuts off an arc on the circle. Ask: how many radii of arc did it cut? That number is the angle's measure in radians.
So one radian is the angle whose arc is exactly one radius long. How big is a full turn? The whole circumference is 2πr — that is 2π radii of arc — so a full turn is 2π radians. A half turn is π radians; a quarter turn (90°) is π/2 radians. The unit is dimensionless: it is a length of arc divided by a length of radius, so the r cancels and the same angle gives the same radian measure on any size of circle.
The radian measure of a central angle is (arc length) ÷ (radius). One full turn = 2π radians because the circumference 2πr is 2π radii of arc. Radians carry no units — they are a pure ratio.
The bridge between the two units is a single equation: a half-turn is 180° and π radians, so
180° = π radians.
Divide both sides and you get the conversion factors. To turn degrees into radians, multiply by π180; to turn radians into degrees, multiply by 180π. For example 60° × π180 = π/3, and 3π/2 × 180π = 270°. These are exactly GX.deg2rad and GX.rad2deg.
The common angles are worth knowing by heart, the way you know your times tables. Notice the pattern: 30°, 45°, 60° are π/6, π/4, π/3 — fractions of π with small denominators — and the quadrantal angles are clean multiples of π/2.
| degrees | 30° | 45° | 60° | 90° | 120° | 135° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|---|
| radians | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | π | 3π/2 | 2π |
Convert 225° to radians: 225 × π180 = 225π180 = 5π/4. Convert 7π/6 to degrees: 7π6 × 180π = 7 × 30° = 210°.
Radians pay off immediately. Back in Stage 18 the arc length of a sector needed a clumsy fraction-of-360° factor. In radians it becomes clean. If a central angle of θ radians cuts an arc on a circle of radius r, then because θ is arc per radius, the arc itself is
s = rθ (θ in radians).
The wedge's area follows by the same logic. A full disk (θ = 2π) has area πr2; the wedge is the fraction θ⁄(2π) of it, so
area = θ2π · πr2 = ½r2θ = ½rs.
The formulas s = rθ and area = ½r2θ hold only when θ is in radians. Plug in degrees and they are simply wrong. A central angle of 90° gives an arc of r·(π/2), not r·90. Convert to radians first, every time.
A lawn sprinkler reaches r = 5 m and sweeps through 120° = 2π/3 radians. Arc watered at the edge: s = 5 · 2π/3 = 10π/3 ≈ 10.47 m. Area watered: ½ · 52 · 2π/3 = 25π/3 ≈ 26.18 m².
An angle is a signed rotation from the positive x-axis: CCW is +, CW is −, and it can run past 360° or below 0°. Adding whole turns keeps the terminal side — those are coterminal angles, sharing every trig value. Measure the turn in radians, where arc ÷ radius is the angle and a full turn is 2π. Carry this table and these two formulas.
| degrees | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| radians | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
180° = π rad · deg→rad ×π180 · rad→deg ×180π · s = rθ · area = ½r2θ (θ in radians)
A ray in standard position is rotated +150°. In which quadrant does its terminal side rest? Then do the same for −150°.
150° is between 90° and 180°, so Quadrant Ⅱ. For −150°, add 360°: −150° + 360° = 210°, which is between 180° and 270°, so Quadrant Ⅲ.
Find one positive and one negative angle coterminal with 100°, and its standard coterminal value in [0°, 360°).
Positive: 100° + 360° = 460°. Negative: 100° − 360° = −260°. Since 100° is already in [0°, 360°), its standard coterminal angle is just 100°.
Convert to radians, as exact π-fractions: 135°, 210°, 300°.
135 × π/180 = 3π/4; 210 × π/180 = 7π/6; 300 × π/180 = 5π/3.
Convert to degrees: 5π/6, 7π/4, 11π/6.
5π/6 × 180/π = 5 × 30° = 150°; 7π/4 × 180/π = 7 × 45° = 315°; 11π/6 × 180/π = 11 × 30° = 330°.
A circle has radius r = 6 cm. A central angle of π/4 radians cuts off an arc and a sector. Find the arc length and the sector area.
Arc: s = rθ = 6 · π/4 = 3π/2 ≈ 4.71 cm. Area: ½r2θ = ½ · 36 · π/4 = 9π/2 ≈ 14.14 cm².
A sector of a circle of radius 10 has arc length s = 25. What is its central angle θ (in radians)? Is that more or less than a quarter turn?
From s = rθ, θ = s/r = 25/10 = 2.5 radians. A quarter turn is π/2 ≈ 1.57 rad and a half turn is π ≈ 3.14 rad, so 2.5 rad is more than a quarter turn but less than a half turn (about 143°).
Six questions to lock it in. Tap the answer you think is right.
This lesson covers CCSS HSF-TF.A.1 (understand radian measure as the ratio of arc length to radius) and HSF-TF.A.2 (extend trig to all real numbers via angles traversed CCW around the unit circle), and revisits arc length and sector area from circle geometry (HSG-C.B.5) now expressed cleanly with s = rθ and area = ½r²θ. The big conceptual shift is treating an angle as a signed rotation rather than a static figure; coterminal angles foreshadow the periodicity studied in 25.4. A common stumbling block is applying s = rθ or ½r²θ with the angle still in degrees — encourage converting to radians as the very first step.