The Quick Answer
The unit circle is a circle with radius 1 centered at the origin. For any angle θ measured counter-clockwise from the positive x-axis, the point on the circle is at (cos θ, sin θ). The x-coordinate gives cosine, the y-coordinate gives sine, and tangent is sin θ / cos θ.
You only need to memorize values for three reference angles — 30°, 45°, and 60°. Every other standard angle is a reflection of one of these into a different quadrant, with the sign determined by which quadrant you land in.
What Is the Unit Circle?
"Unit" means 1. The unit circle is a circle with a radius of exactly 1, drawn on a standard x-y coordinate plane with its center at (0, 0).
It exists to answer one question: given an angle, what are the sine and cosine?
On any circle of radius r, a point at angle θ sits at (r·cos θ, r·sin θ). When r = 1, the coordinates simplify to (cos θ, sin θ). No multiplication needed — just read the coordinates.
This is why trigonometry courses spend so much time on it. The unit circle is the definition of sine and cosine for all angles, not just acute ones that fit in a right triangle.
The Three Angles You Actually Need to Know
Most of the unit circle comes from just three right triangles inscribed in the circle:
The 30-60-90 Triangle
A right triangle with angles 30°, 60°, and 90° has sides in the ratio 1 : √3 : 2. Scale it so the hypotenuse (the radius) equals 1:
- 30° (π/6): sin = 1/2, cos = √3/2
- 60° (π/3): sin = √3/2, cos = 1/2
Notice the swap: the sine and cosine values at 30° and 60° are the same pair, just reversed.
The 45-45-90 Triangle
A right triangle with two 45° angles has sides in the ratio 1 : 1 : √2. Scale it:
- 45° (π/4): sin = √2/2, cos = √2/2
Both coordinates are equal, because the angle bisects the first quadrant exactly.
The Axis Points
The four axis angles are straightforward:
- 0° (0): (1, 0) — sin = 0, cos = 1
- 90° (π/2): (0, 1) — sin = 1, cos = 0
- 180° (π): (−1, 0) — sin = 0, cos = −1
- 270° (3π/2): (0, −1) — sin = −1, cos = 0
These use no triangles at all — they are just the four compass points of the circle.
How to Get Every Other Standard Angle
The remaining 12 standard angles (120°, 135°, 150°, 210°, 225°, 240°, 300°, 315°, 330°) are all reflections of 30°, 45°, or 60° into quadrants II, III, or IV.
The process:
- Find the reference angle — the acute angle between the terminal side and the x-axis.
- Look up the sine and cosine for the reference angle.
- Apply the signs for the target quadrant.
Example: sin 225°
225° is in Quadrant III (between 180° and 270°).
- Reference angle: 225° − 180° = 45°
- sin 45° = √2/2
- In Q3, sine is negative
- sin 225° = −√2/2
Example: cos 300°
300° is in Quadrant IV (between 270° and 360°).
- Reference angle: 360° − 300° = 60°
- cos 60° = 1/2
- In Q4, cosine is positive
- cos 300° = 1/2
Quadrant Signs: "All Students Take Calculus"
The mnemonic tells you which trig functions are positive in each quadrant:
- Quadrant I (0°–90°): All positive
- Quadrant II (90°–180°): Sine positive (cos and tan negative)
- Quadrant III (180°–270°): Tangent positive (sin and cos negative)
- Quadrant IV (270°–360°): Cosine positive (sin and tan negative)
The logic behind it: in Q1, both x and y are positive, so sin (+) and cos (+). In Q2, x is negative but y is positive, so cos (−) and sin (+). And so on.
Tangent on the Unit Circle
Tangent is always sin θ divided by cos θ:
tan θ = y / x
This means:
- At 45°, tan = (√2/2) / (√2/2) = 1
- At 60°, tan = (√3/2) / (1/2) = √3
- At 90°, tan = 1 / 0 = undefined (division by zero)
Tangent is undefined at 90° and 270° because those points sit on the y-axis where x = 0.
Degrees and Radians
Degrees and radians are two units for measuring the same thing: the size of an angle.
Conversion formula: radians = degrees × π / 180
Key equivalences:
| Degrees | Radians |
|---|---|
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
The pattern: denominators of 6, 4, and 3 correspond to the three key reference angles.
Memorization Tips
The "Hand Trick"
Hold up your left hand, fingers spread. Assign angles to your fingers from left to right: 90°, 60°, 45°, 30°, 0°.
For sine: count the fingers above the one you fold down, take the square root, and divide by 2.
- 0° → 0 fingers above → √0/2 = 0
- 30° → 1 finger above → √1/2 = 1/2
- 45° → 2 fingers above → √2/2
- 60° → 3 fingers above → √3/2
- 90° → 4 fingers above → √4/2 = 1
For cosine, reverse the order (count fingers below).
The Pattern in the Numbers
Write the sine values for 0° through 90°:
0, 1/2, √2/2, √3/2, 1
Under the radical, the numbers go: 0, 1, 2, 3, 4 — then divide by 2. That is the entire pattern.
Cosine is the same sequence in reverse order.
Common Mistakes
Swapping sine and cosine. Sine is the y-coordinate (vertical). Cosine is the x-coordinate (horizontal). A useful mnemonic: "x before y" alphabetically, "cosine before sine" alphabetically.
Wrong signs in Quadrants III and IV. At 210°, both sin and cos are negative. At 330°, sine is negative but cosine is positive. Always check the quadrant.
Calculator in wrong mode. If your calculator returns sin(90) ≈ 0.894, it is in radian mode. Switch to degrees, or use sin(π/2) instead.
Saying tan 90° = 0. It is undefined, not zero. The function blows up to ±∞ near 90°.
Confusing √2 and √3 values. At 45°, both sine and cosine are √2/2 (≈ 0.707). At 30° and 60°, the values are 1/2 (0.5) and √3/2 (≈ 0.866).
Worked Examples
Example 1: Find all trig values at 150°
150° is in Quadrant II. Reference angle = 180° − 150° = 30°.
- sin 150° = +sin 30° = 1/2 (sine is positive in Q2)
- cos 150° = −cos 30° = −√3/2 (cosine is negative in Q2)
- tan 150° = sin/cos = (1/2) / (−√3/2) = −√3/3 (equivalently −1/√3)
Example 2: Find cos 240°
240° is in Quadrant III. Reference angle = 240° − 180° = 60°.
- cos 60° = 1/2
- In Q3, cosine is negative
- cos 240° = −1/2
Example 3: Is sin 315° positive or negative?
315° is in Quadrant IV. Sine is negative in Q4.
Reference angle = 360° − 315° = 45°.
- sin 315° = −√2/2
Where the Unit Circle Shows Up
The unit circle is not just a classroom exercise. It appears anywhere angles or periodic behavior matters:
- Physics: Circular motion, wave equations, pendulum oscillation, and projectile trajectories all use sine and cosine functions defined by the unit circle.
- Engineering: AC circuit analysis relies on sinusoidal functions. Signal processing uses Fourier transforms built from sine and cosine waves.
- Computer graphics: Rotating objects on screen requires multiplying coordinates by sin θ and cos θ — the exact values the unit circle provides.
- Music and acoustics: Sound waves are modeled as combinations of sine waves at different frequencies.
- Navigation: Converting between polar coordinates (bearing + distance) and Cartesian coordinates (x, y) uses the unit circle relationship directly.
Try It Yourself
Use the interactive unit circle tool to click through angles and see the sine, cosine, and tangent values update in real time. The dashed projection lines show exactly how the coordinates map to trig values.
FAQ
How do I remember which values go with which angle?
The sine values from 0° to 90° follow the pattern √0/2, √1/2, √2/2, √3/2, √4/2 — the number under the radical goes 0 through 4. Cosine is the same sequence reversed.
Can I use the unit circle for negative angles?
Yes. A negative angle rotates clockwise. −30° lands at the same spot as 330° and has the same trig values.
What about angles larger than 360°?
They wrap around. Subtract 360° until you get an angle between 0° and 360°. For example, 750° − 360° − 360° = 30°, so sin 750° = sin 30° = 1/2.
Why do textbooks use radians instead of degrees?
Radians make calculus cleaner. The derivative of sin(x) is cos(x) only when x is in radians. In degrees, you'd need an extra conversion factor of π/180.
What is secant, cosecant, and cotangent on the unit circle?
They are the reciprocals: sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = cos θ/sin θ. Each is undefined where its "partner" is zero.
Is there a unit circle for hyperbolic functions?
Not exactly. Hyperbolic functions (sinh, cosh) relate to the unit hyperbola (x² − y² = 1) rather than the unit circle (x² + y² = 1). The geometry is analogous but the curve is different.
How accurate are the decimal values on the chart?
The exact values (like √2/2) are precise. The decimals are rounded to 4 places. √2/2 ≈ 0.7071, √3/2 ≈ 0.8660, and √3/3 ≈ 0.5774. For most practical purposes, 4 decimal places are sufficient.
What is the Pythagorean identity?
For any angle θ on the unit circle: sin²θ + cos²θ = 1. This comes directly from the equation of the circle (x² + y² = 1) with x = cos θ and y = sin θ.