All 16 standard angles with exact sin, cos, and tan values — plus the memory tricks, ASTC rule, and reference angle method that make it stick.
The unit circle is a circle with radius 1 centered at the origin. For any angle θ measured counterclockwise from the positive x-axis, the terminal side of the angle intersects the circle at the point:
(x, y) = (cos θ, sin θ)
x = cos θ | y = sin θ | tan θ = sin θ / cos θ = y / x
Because the radius is 1, the coordinates of every point on the circle are exactly the trig values — no scaling needed. This is why the unit circle is the foundation for all of trigonometry: it converts angles directly into coordinates.
Key equation: x² + y² = 1
Since every point satisfies the circle equation, we get cos²θ + sin²θ = 1 for free — the most important Pythagorean identity.
Color-coded by quadrant:QIQIIQIIIQIV
| Degrees | Radians | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
| 120° | 2π/3 | −1/2 | √3/2 | −√3 |
| 135° | 3π/4 | −√2/2 | √2/2 | −1 |
| 150° | 5π/6 | −√3/2 | 1/2 | −√3/3 |
| 180° | π | −1 | 0 | 0 |
| 210° | 7π/6 | −√3/2 | −1/2 | √3/3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 |
| 240° | 4π/3 | −1/2 | −√3/2 | √3 |
| 270° | 3π/2 | 0 | −1 | undefined |
| 300° | 5π/3 | 1/2 | −√3/2 | −√3 |
| 315° | 7π/4 | √2/2 | −√2/2 | −1 |
| 330° | 11π/6 | √3/2 | −1/2 | −√3/3 |
Note: √3/3 is the simplified form of 1/√3 (rationalized denominator). Some textbooks write tan 30° = 1/√3.
As the angle increases from 0° to 90°, the numerator for sin follows the sequence 0, 1, √2, √3, 2. All values are divided by 2 (except 0 and 1 at the endpoints).
| Angle | sin θ | cos θ | Numerator pattern |
|---|---|---|---|
| 0° | 0 | 1 | sin: 0 → cos: 2 |
| 30° | 1/2 | √3/2 | sin: 1 → cos: √3 |
| 45° | √2/2 | √2/2 | sin: √2 → cos: √2 |
| 60° | √3/2 | 1/2 | sin: √3 → cos: 1 |
| 90° | 1 | 0 | sin: 2 → cos: 0 |
The shortcut
Write the sequence 0, 1, 2, 3, 4 under the angles 0°, 30°, 45°, 60°, 90°. Take the square root of each: √0, √1, √2, √3, √4. Divide by 2. That gives sin. Read it backwards for cos.
Once you know QI values, ASTC tells you the sign for every other quadrant. Only the listed function (and its reciprocal) is positive; the others are negative.
Quadrant I (0°–90°)
All positive
sin, cos, tan all +
Quadrant II (90°–180°)
Sine positive
sin +, cos −, tan −
Quadrant III (180°–270°)
Tangent positive
sin −, cos −, tan +
Quadrant IV (270°–360°)
Cosine positive
sin −, cos +, tan −
Example: sin(210°)
210° is in QIII. Reference angle = 210° − 180° = 30°. In QIII, sine is negative. So sin(210°) = −sin(30°) = −1/2.
A reference angle is the positive acute angle (between 0° and 90°) formed between the terminal side of your angle and the x-axis. The trig values of any angle equal the trig values of its reference angle — with signs determined by ASTC.
| Quadrant | Angle range | Reference angle formula | Example |
|---|---|---|---|
| QI | 0° to 90° | θ′ = θ | θ = 60° → θ′ = 60° |
| QII | 90° to 180° | θ′ = 180° − θ | θ = 150° → θ′ = 30° |
| QIII | 180° to 270° | θ′ = θ − 180° | θ = 225° → θ′ = 45° |
| QIV | 270° to 360° | θ′ = 360° − θ | θ = 300° → θ′ = 60° |
In radians: same idea
QII: θ′ = π − θ
QIII: θ′ = θ − π
QIV: θ′ = 2π − θ
Sides in ratio 1 : √3 : 2
hypotenuse = 2, opposite 30° = 1, opposite 60° = √3
sin 30° = 1/2 | sin 60° = √3/2
cos 30° = √3/2 | cos 60° = 1/2
tan 30° = √3/3 | tan 60° = √3
Origin: an equilateral triangle with side 2 split in half. The half gives you sides 1, 2, and √3 (by Pythagoras: 2² − 1² = 3, so the third side is √3).
Sides in ratio 1 : 1 : √2
hypotenuse = √2, both legs = 1
sin 45° = √2/2
cos 45° = √2/2
tan 45° = 1
Origin: a square with side 1 cut diagonally. The diagonal (hypotenuse) is √(1² + 1²) = √2. Both acute angles are 45° by symmetry.
Why these triangles matter
Every angle in the unit circle table is built from one of these two triangles. The 30° and 60° values come from the 30-60-90 triangle; the 45° value comes from the 45-45-90 triangle. For 0°, 90°, 180°, 270°, the values are just 0 and 1 from the axes. If you can reconstruct these two triangles, you can derive the entire unit circle from scratch.
Inverse trig functions reverse the process: given a value, find the angle. Because trig functions are not one-to-one, their inverses require restricted domains.
| Function | Notation | Input range | Output range | Quadrants used |
|---|---|---|---|---|
| Arcsine | sin⁻¹(x) or arcsin(x) | [−1, 1] | [−π/2, π/2] | QI and QIV |
| Arccosine | cos⁻¹(x) or arccos(x) | [−1, 1] | [0, π] | QI and QII |
| Arctangent | tan⁻¹(x) or arctan(x) | (−∞, ∞) | (−π/2, π/2) | QI and QIV |
Common mistake
sin⁻¹(sin(150°)) ≠ 150°. The output of arcsin is restricted to [−π/2, π/2], so arcsin(sin(150°)) = arcsin(1/2) = 30°. Always check which quadrant the output falls in for the given inverse function.
Find arcsin(−√2/2)
We need an angle in [−π/2, π/2] whose sine is −√2/2.
sin(45°) = √2/2, so the reference angle is 45°.
Since the output must be in [−π/2, π/2] and sin is negative, use the negative angle.
arcsin(−√2/2) = −45° = −π/4
Find arccos(−1/2)
We need an angle in [0, π] whose cosine is −1/2.
cos(60°) = 1/2, so the reference angle is 60°.
In [0, π], cosine is negative in QII, so the angle is 180° − 60° = 120°.
arccos(−1/2) = 120° = 2π/3
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. For any angle θ measured from the positive x-axis, the point where the terminal side of the angle intersects the circle has coordinates (cos θ, sin θ). This means every point on the unit circle directly gives you the cosine (x-coordinate) and sine (y-coordinate) of that angle. Tangent is then sin θ / cos θ = y/x.
Use the 1-√2-√3 pattern: as the angle increases from 0° to 90°, the numerators for sin go 0, 1, √2, √3, 2 (all over 2, except 0 and 1). For cos, the numerators go in reverse: 2, √3, √2, 1, 0. So at 0°: sin=0, cos=1. At 30°: sin=1/2, cos=√3/2. At 45°: sin=√2/2, cos=√2/2. At 60°: sin=√3/2, cos=1/2. At 90°: sin=1, cos=0. For angles beyond 90°, apply the ASTC rule to determine signs and use the reference angle to get the magnitude.
ASTC stands for All Students Take Calculus and tells you which trig functions are positive in each quadrant. Quadrant I (0°–90°): All functions positive. Quadrant II (90°–180°): Sine (and cosecant) positive. Quadrant III (180°–270°): Tangent (and cotangent) positive. Quadrant IV (270°–360°): Cosine (and secant) positive. To find sin(150°): the reference angle is 30°, and in QII only sin is positive, so sin(150°) = +sin(30°) = 1/2.
Flashcards, fill-in-the-table drills, and timed quizzes — until you can write the entire unit circle from memory.
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