A complete precalculus guide: what the unit circle is, why it works, how to read it, and every trick you need to memorize it fast. Covers all 16 key angles, ASTC, reference angles, special triangles, and the graphs of sine and cosine.
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Its equation is:
x² + y² = 1
For any angle θ measured counterclockwise from the positive x-axis, the terminal side of the angle intersects the circle at a point whose coordinates are:
(x, y) = (cos θ, sin θ)
x = cos θ | y = sin θ | tan θ = y / x
Because the radius is exactly 1, there is no scaling factor. The x-coordinate IS the cosine and the y-coordinate IS the sine. This is the key insight: the unit circle turns angles into coordinates and coordinates back into angles.
Why radius 1?
For a circle of radius r, the point on the circle at angle θ would be (r cos θ, r sin θ). When r = 1, that simplifies to (cos θ, sin θ) directly. Radius 1 eliminates the scaling factor and makes trig values read off the circle as plain coordinates.
An angle is in standard position when:
Positive angles
Measured counterclockwise from the initial side. 90° points straight up.
Negative angles
Measured clockwise. −90° points straight down, same as +270°.
Two angles are coterminal if their terminal sides land in the same place. Add or subtract 360° (or 2π radians) repeatedly to find coterminal angles.
30° and 390° are coterminal → 390 = 30 + 360
30° and −330° are coterminal → −330 + 360 = 30
π/3 and π/3 + 2π = 7π/3 are coterminal
Coterminal angles always have identical sin, cos, and tan values because they hit the same point on the unit circle.
Degrees are the everyday measure (a full circle = 360°). Radians are the mathematically natural measure: one radian is the angle that cuts an arc equal in length to the radius. Since the circumference of a unit circle is 2π, a full rotation is 2π radians.
180° = π radians
To convert: multiply degrees × (π / 180) for radians, or multiply radians × (180 / π) for degrees.
| Degrees | Radians | Decimal approx. |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 120° | 2π/3 | 2.0944 |
| 135° | 3π/4 | 2.3562 |
| 150° | 5π/6 | 2.6180 |
| 180° | π | 3.1416 |
| 270° | 3π/2 | 4.7124 |
| 360° | 2π | 6.2832 |
Color-coded by quadrant:QIQIIQIIIQIVAxes
| Degrees | Radians | Quadrant | cos θ | sin θ | tan θ |
|---|---|---|---|---|---|
| 0° | 0 | — | 1 | 0 | 0 |
| 30° | π/6 | I | √3/2 | 1/2 | √3/3 |
| 45° | π/4 | I | √2/2 | √2/2 | 1 |
| 60° | π/3 | I | 1/2 | √3/2 | √3 |
| 90° | π/2 | — | 0 | 1 | undefined |
| 120° | 2π/3 | II | −1/2 | √3/2 | −√3 |
| 135° | 3π/4 | II | −√2/2 | √2/2 | −1 |
| 150° | 5π/6 | II | −√3/2 | 1/2 | −√3/3 |
| 180° | π | — | −1 | 0 | 0 |
| 210° | 7π/6 | III | −√3/2 | −1/2 | √3/3 |
| 225° | 5π/4 | III | −√2/2 | −√2/2 | 1 |
| 240° | 4π/3 | III | −1/2 | −√3/2 | √3 |
| 270° | 3π/2 | — | 0 | −1 | undefined |
| 300° | 5π/3 | IV | 1/2 | −√3/2 | −√3 |
| 315° | 7π/4 | IV | √2/2 | −√2/2 | −1 |
| 330° | 11π/6 | IV | √3/2 | −1/2 | −√3/3 |
Note: √3/3 is the rationalized form of 1/√3. Some textbooks write tan 30° = 1/√3. Both are correct.
In Quadrant I (0° to 90°), the numerator for sin increases as: 0, 1, √2, √3, 2. All divided by 2 (though 0/2 = 0 and 2/2 = 1). For cos, the numerators go in reverse: 2, √3, √2, 1, 0.
0°
sin: 0
cos: 1
30°
sin: 1/2
cos: √3/2
45°
sin: √2/2
cos: √2/2
60°
sin: √3/2
cos: 1/2
90°
sin: 1
cos: 0
The unit circle values are not arbitrary — they come directly from two special right triangles. Memorize these triangles and you can reconstruct the entire Quadrant I of the unit circle from scratch.
Side ratios (hypotenuse = 1)
Short leg (opposite 30°) = 1/2
Long leg (opposite 60°) = √3/2
Hypotenuse = 1
sin(30°) = opposite/hyp = 1/2
cos(30°) = adjacent/hyp = √3/2
sin(60°) = opposite/hyp = √3/2
cos(60°) = adjacent/hyp = 1/2
Generating rule
Start with any 30-60-90 triangle. Multiply or divide sides by any constant. The ratios (1 : √3 : 2) stay the same. The unit circle version just sets hyp = 1.
Side ratios (hypotenuse = 1)
Leg 1 (opposite 45°) = √2/2
Leg 2 (opposite 45°) = √2/2
Hypotenuse = 1
sin(45°) = √2/2
cos(45°) = √2/2
tan(45°) = 1 (legs are equal)
Generating rule
Start with a right isosceles triangle (two equal legs). The hypotenuse is always leg × √2. The unit circle version sets hyp = 1, so leg = 1/√2 = √2/2.
How to reconstruct the full unit circle
The reference angle for any angle θ is the acute angle (between 0° and 90°) formed between the terminal side and the nearest part of the x-axis. Reference angles let you use QI values everywhere — just attach the right sign afterward.
QI (0° to 90°)
Reference angle = θ itself
θ = 55° → ref = 55°
QII (90° to 180°)
Reference angle = 180° − θ
θ = 150° → ref = 30°
QIII (180° to 270°)
Reference angle = θ − 180°
θ = 210° → ref = 30°
QIV (270° to 360°)
Reference angle = 360° − θ
θ = 315° → ref = 45°
Worked Example
Find sin(330°). Step 1: 330° is in QIV. Step 2: reference angle = 360° − 330° = 30°. Step 3: sin(30°) = 1/2. Step 4: In QIV, sine is negative (ASTC: only cosine is positive). Therefore sin(330°) = −1/2.
For radians
Same idea, just substitute π for 180° and 2π for 360°. Example: reference angle for 5π/6 (QII) is π − 5π/6 = π/6 (= 30°).
The mnemonic ASTC — All Students Take Calculus — tells you which of sin, cos, and tan are positive in each quadrant. All others in that quadrant are negative.
QI: All positive
sin > 0, cos > 0, tan > 0
0° to 90°
QII: Sine positive
sin > 0, cos < 0, tan < 0
90° to 180°
QIII: Tangent positive
sin < 0, cos < 0, tan > 0
180° to 270°
QIV: Cosine positive
sin < 0, cos > 0, tan < 0
270° to 360°
Because cos θ = x and sin θ = y. In QI, both x and y are positive, so sin and cos are positive, and tan = y/x is positive too. In QII, x is negative and y is positive, so cos is negative, sin is positive, and tan (negative over positive) is negative. In QIII, both x and y are negative, so sin and cos are negative, but tan (negative over negative) is positive. In QIV, x is positive and y is negative, so cos is positive and sin and tan are negative.
Worked Examples with ASTC
tan(225°): ref angle = 45°, tan(45°) = 1, QIII → tan positive → tan(225°) = +1
cos(120°): ref angle = 60°, cos(60°) = 1/2, QII → cos negative → cos(120°) = −1/2
sin(300°): ref angle = 60°, sin(60°) = √3/2, QIV → sin negative → sin(300°) = −√3/2
sin²θ + cos²θ = 1
The foundation
1 + tan²θ = sec²θ
Divide by cos²θ
cot²θ + 1 = csc²θ
Divide by sin²θ
Proof of the main identity
Every point on the unit circle satisfies x² + y² = 1. Substitute x = cos θ and y = sin θ and you immediately get cos²θ + sin²θ = 1. No derivation required — it IS the circle equation.
csc θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ = cos θ / sin θ
csc is undefined when sin = 0 (at 0°, 180°, 360°). sec is undefined when cos = 0 (at 90°, 270°). cot is undefined when tan is undefined or when sin = 0.
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
These come directly from the unit circle: tan θ = y/x = sin θ/cos θ.
The unit circle traces out the graphs of sine and cosine. As the angle θ increases from 0 to 2π, the y-coordinate (sin) traces a wave starting at 0, rising to 1, falling to −1, and returning to 0. The x-coordinate (cos) traces the same wave but starting at 1 instead.
| Feature | y = sin(x) | y = cos(x) |
|---|---|---|
| Amplitude | 1 | 1 |
| Period | 2π | 2π |
| Domain | all reals | all reals |
| Range | [−1, 1] | [−1, 1] |
| Starting value | sin(0) = 0 | cos(0) = 1 |
| Zeros | 0, π, 2π, … | π/2, 3π/2, … |
| Maximum | 1 at π/2 | 1 at 0 |
| Minimum | −1 at 3π/2 | −1 at π |
| Even/Odd | Odd: sin(−x) = −sin(x) | Even: cos(−x) = cos(x) |
The general form is y = A • sin(Bx + C) + D (same for cosine). Each parameter shifts or scales the graph:
A
Amplitude
|A| stretches vertically. A < 0 flips the graph upside down.
B
Period
Period = 2π / |B|. Larger B compresses the graph horizontally.
C
Phase shift
Shift = −C/B. Positive shift is left, negative is right.
D
Vertical shift
Midline moves to y = D. Shifts the entire wave up or down.
Example
y = 3 • sin(2x − π) + 1
A = 3 → amplitude 3
B = 2 → period = 2π/2 = π
C = −π → phase shift = −(−π)/2 = π/2 (right)
D = 1 → midline at y = 1, range [−2, 4]
The tangent graph is very different from sine and cosine. It has period π (not 2π) and has vertical asymptotes wherever cos θ = 0 (at π/2, 3π/2, etc.) because tan = sin/cos. Between asymptotes, tan increases from −∞ to +∞. There is no amplitude because the range is all real numbers.
Any object moving in a circle of radius r has position (r·cosθ, r·sinθ). The unit circle is the special case r = 1. Ferris wheels, gears, satellite orbits — all use this formula.
Sound pressure, light intensity, and radio signals are all modeled as y = A·sin(2πft + φ) where f is frequency and φ is phase. The unit circle defines what “sine” means in those formulas.
Household electricity is not constant — it oscillates sinusoidally at 60 Hz in the US. Electrical engineers use the unit circle constantly when analyzing circuits with AC voltage.
Converting compass bearing angles to east-west and north-south displacements uses cos and sin. GPS satellites use spherical trigonometry built on the same foundations.
Any periodic signal — music, heartbeats, seismic waves — can be broken into sine and cosine components at different frequencies. This is Fourier analysis, and it runs on unit circle math.
Rotating objects in 2D and 3D games uses rotation matrices built from cos and sin of the rotation angle. Every time something spins on screen, the unit circle is doing the work.
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. For any angle measured counterclockwise from the positive x-axis, the terminal side intersects the circle at the point (cos θ, sin θ).
Because the radius is exactly 1, the x-coordinate IS the cosine and the y-coordinate IS the sine — no scaling needed. This is why the unit circle is the foundation of all trigonometry: every angle maps directly to a point, and that point gives you the trig values.
Multiply by π/180 to go degrees → radians. Multiply by 180/π to go radians → degrees.
The anchor fact: 180° = π radians. Everything else follows from that.
Examples: 60° × (π/180) = π/3. And π/4 × (180/π) = 45°.
Coterminal angles share the same terminal side. Add or subtract 360° (or 2π radians) to find them.
Example: 30° and 390° are coterminal. So are 30° and −30° + 360° = −330°. All coterminal angles have identical sin, cos, and tan values.
A reference angle is the acute angle (0° to 90°) between the terminal side and the nearest x-axis.
QI: reference angle = θ itself. QII: 180° − θ. QIII: θ − 180°. QIV: 360° − θ.
Example: reference angle for 210° is 210° − 180° = 30°. Then use ASTC to find the correct sign.
ASTC = All Students Take Calculus. It tells you which trig functions are positive by quadrant.
QI: All positive. QII: Sine positive. QIII: Tangent positive. QIV: Cosine positive.
Worked example: cos(240°). Reference angle = 60°, cos(60°) = 1/2. QIII → cosine is negative. Answer: cos(240°) = −1/2.
They come from special right triangles placed inside the circle.
30-60-90 triangle with hypotenuse 1: legs are 1/2 and √3/2. So sin(30°) = 1/2, cos(30°) = √3/2, sin(60°) = √3/2, cos(60°) = 1/2.
45-45-90 triangle with hypotenuse 1: both legs are √2/2. So sin(45°) = cos(45°) = √2/2.
Tiling these triangles in all four quadrants (with sign changes from ASTC) fills in all 16 standard angles.
sin²θ + cos²θ = 1. It comes from the unit circle equation x² + y² = 1, with x = cosθ and y = sinθ.
Two derived forms: divide by cos²θ to get 1 + tan²θ = sec²θ. Divide by sin²θ to get cot²θ + 1 = csc²θ.
You can use this to find sin if you know cos, or to simplify trig expressions on exams.
Use the 1–√2–√3 pattern. As the angle goes 0° → 30° → 45° → 60° → 90°, the numerator of sin goes: 0, 1, √2, √3, 2. All over 2.
Cosine reverses: 2, √3, √2, 1, 0. All over 2.
tan = sin/cos. tan(30°) = (1/2)/(√3/2) = 1/√3 = √3/3. tan(45°) = 1. tan(60°) = √3.
cscθ = 1/sinθ. secθ = 1/cosθ. cotθ = 1/tanθ = cosθ/sinθ.
csc is undefined when sin = 0 (at 0° and 180°). sec is undefined when cos = 0 (at 90° and 270°). cot is undefined when sin = 0.
Memory tip: co-functions pair up. cosecant pairs with sine, cotangent with tangent, secant with cosine.
For y = A·sin(Bx + C) + D: amplitude = |A|, period = 2π/|B|, phase shift = −C/B, vertical shift = D.
y = sin(x) has amplitude 1, period 2π, no shift.
y = 3·sin(2x − π) + 1 has amplitude 3, period π, phase shift +π/2 (right), vertical shift +1.
Cosine has the same period and amplitude rules. Tangent has period π, not 2π.
Circular motion: the position of any point on a rotating wheel of radius r is (r·cosθ, r·sinθ).
Waves: sound, light, and AC electricity are all modeled with sine and cosine functions.
Navigation: compass bearings are converted to x-y displacements using cos and sin.
Signal processing: Fourier analysis decomposes any complex wave into sine and cosine components. All of this rests on the unit circle.
Standard position means the vertex is at the origin and the initial side lies along the positive x-axis.
Angles measured counterclockwise are positive; clockwise is negative.
Example: −90° has its terminal side pointing straight down (negative y-axis), same position as +270°. They are coterminal.
x² + y² = 1
(x, y) = (cos θ, sin θ)
tan θ = y / x
deg → rad: × (π / 180)
rad → deg: × (180 / π)
180° = π rad
QI: All +
QII: Sin +
QIII: Tan +
QIV: Cos +
QI: θ
QII: 180° − θ
QIII: θ − 180°
QIV: 360° − θ
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
cot²θ + 1 = csc²θ
y = A sin(Bx + C) + D
Amp = |A|
Period = 2π / |B|
Shift = −C / B
Precalculus Study Guide
Full roadmap for precalculus
Trig Identities
Sum, difference, double-angle, and more
Right Triangle Trig
SOH-CAH-TOA and triangle applications
Solving Trig Equations
Find all solutions using the unit circle
Trig Graphs
Sine, cosine, tangent graph sketching
Inverse Trig Functions
arcsin, arccos, arctan domains and ranges
Law of Sines and Cosines
Oblique triangle solving
Polar Coordinates
r and theta coordinate system
Parametric Equations
Circular and elliptical parametric curves
Knowing the unit circle is one thing. Being able to pull the right value under exam pressure is another. NailTheTest builds adaptive quizzes that drill you on exactly the angles and quadrants you keep missing — until the unit circle is automatic.