Trigonometric Functions and Reference Angles
Enter any angle to see its sine, cosine, tangent, and cotangent, exact values for special angles, decimals otherwise, with the sign explained by quadrant.
What are trigonometric functions?
Sine, cosine, tangent, cotangent, secant, and cosecant are the six trigonometric functions. Each relates an angle to a ratio of the coordinates of a point on the unit circle: sine is the y-coordinate, cosine is the x-coordinate, tangent is sine divided by cosine, and cotangent, secant, and cosecant are their respective reciprocals.
For any angle θ at point (x, y) on the unit circle: sin θ = y, cos θ = x, tan θ = y/x, cot θ = x/y, sec θ = 1/x, and csc θ = 1/y. Because x and y can be positive or negative depending on the quadrant, each function's sign changes predictably from quadrant to quadrant.
These same functions have an older definition, too, based on right triangles rather than the unit circle: sine is the ratio of the opposite side to the hypotenuse, cosine is the adjacent side to the hypotenuse, and tangent is the opposite side to the adjacent side. That definition only works for acute angles inside a triangle, capped at 90°. The unit circle definition is the generalization that keeps working past 90°, through negative angles, and around as many full turns as needed. The right-triangle ratios and the unit circle coordinates agree perfectly for any angle between 0° and 90°, which is exactly why the reference angle approach works: it translates a difficult angle back into the simple right-triangle case.
Secant and cosecant are less commonly used than sine, cosine, and tangent, but they follow the same reference angle rules. Since secant is the reciprocal of cosine, it's positive exactly where cosine is positive, Quadrants I and IV. Cosecant, the reciprocal of sine, is positive where sine is positive, Quadrants I and II. Reciprocal functions are undefined wherever the function they invert equals zero, which is why secant is undefined at 90° and 270°, and cosecant is undefined at 0°, 180°, and 360°.
The Reference Angle Theorem
sin θ = ±sin θ̂, cos θ = ±cos θ̂, tan θ = ±tan θ̂
Every trig function of an angle θ equals the same function of its reference angle θ̂, up to sign. For example, 210° has reference angle 30°. Since sin(30°) = 1/2 and sine is negative in Quadrant III (where 210° lives), sin(210°) = −1/2, same magnitude, flipped sign.
This theorem is what makes trigonometry practical without a calculator for every problem. There are infinitely many angles, but only a handful of distinct reference angles show up in typical coursework: 0°, 30°, 45°, 60°, and 90°. Memorize the sine, cosine, and tangent of those five values once, memorize the ASTC sign pattern, and every angle that reduces to one of them, which is most angles used in textbooks and exams, becomes solvable from memory alone.
The theorem itself comes from a shape called the reference triangle: drop a perpendicular line from any point on the terminal side straight down to the x-axis, and the right triangle formed that way always has the reference angle as one of its acute angles. That triangle's side lengths give the same sine, cosine, and tangent ratios in every quadrant, which is exactly why the Reference Angle Theorem holds regardless of where the terminal side points.
The theorem also extends automatically to coterminal angles: sin(θ + 360°n) = sin(θ) for any integer n, and the same is true of cosine and tangent. This periodicity, sometimes called rotational symmetry, is why adding or subtracting full turns never changes a trig value, and it's the reason coterminal angles always share the same reference angle in the first place.
Sign rules per quadrant
| Function | Q I | Q II | Q III | Q IV |
|---|---|---|---|---|
| sin | + | + | − | − |
| cos | + | − | − | + |
| tan | + | − | + | − |
| cot | + | − | + | − |
Exact trig values for special angles
These seventeen angles cover every standard position from 0° to 360° at 15°, 30°, and 45° boundaries. Notice that the magnitudes repeat in blocks of three, 1/2, √2/2, and √3/2, while only the signs shift between quadrants, exactly as the Reference Angle Theorem predicts.
Cotangent is included alongside sine, cosine, and tangent because it comes up almost as often in coursework, and it's easy to derive once the others are known: cot θ = cos θ / sin θ, the reciprocal of tangent. Wherever tangent is undefined, cotangent equals 0, and wherever tangent equals 0, cotangent is undefined. The two functions trade off exactly at the quadrantal angles.
A practical shortcut worth knowing: tangent and cotangent share the same sign in every quadrant, since one is just the reciprocal of the other and dividing by a positive or negative number never flips a sign on its own. That's why the sign table above shows identical +/− patterns for tan and cot across all four quadrants, while sine and cosine each follow their own separate pattern.
| Angle (°) | Angle (rad) | sin | cos | tan | cot |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | - |
| 30° | π/6 | 1/2 | √3/2 | √3/3 | √3 |
| 45° | π/4 | √2/2 | √2/2 | 1 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 | √3/3 |
| 90° | π/2 | 1 | 0 | - | 0 |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 | −√3/3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −√3/3 | −√3 |
| 180° | π | 0 | −1 | 0 | - |
| 210° | 7π/6 | −1/2 | −√3/2 | √3/3 | √3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 | 1 |
| 240° | 4π/3 | −√3/2 | −1/2 | √3 | √3/3 |
| 270° | 3π/2 | −1 | 0 | - | 0 |
| 300° | 5π/3 | −√3/2 | 1/2 | −√3 | −√3/3 |
| 315° | 7π/4 | −√2/2 | √2/2 | −1 | −1 |
| 330° | 11π/6 | −1/2 | √3/2 | −√3/3 | −√3 |
| 360° | 2π | 0 | 1 | 0 | - |
Frequently Asked Questions
What is the sine of a reference angle?
The sine of a reference angle equals the absolute value of the sine of the original angle. Only the sign can differ, and that sign is determined by the quadrant of the original angle. This same rule applies equally to cosine, tangent, and their reciprocal functions.
How do reference angles help find trig values?
Reference angles reduce any angle to an equivalent acute angle between 0° and 90°. Since trig functions repeat their magnitude at every reference angle, you only need to memorize first-quadrant values and then apply the correct sign for the quadrant.
What is sin(150°)?
sin(150°) = 1/2. 150° is in Quadrant II with reference angle 30°, and sine is positive in Quadrant II, so sin(150°) = sin(30°) = 1/2. Cosine and tangent are both negative at 150°, since only sine stays positive in Quadrant II.
What is cos(225°)?
cos(225°) = −√2/2. 225° is in Quadrant III with reference angle 45°, and cosine is negative in Quadrant III, so cos(225°) = −cos(45°) = −√2/2. Sine is also negative at 225°, while tangent alone stays positive throughout all of Quadrant III.
What is tan(300°)?
tan(300°) = −√3. 300° is in Quadrant IV with reference angle 60°, and tangent is negative in Quadrant IV, so tan(300°) = −tan(60°) = −√3. Cosine stays positive at 300° as well, while sine alone is negative throughout Quadrant IV.
How do inverse trig functions relate to reference angles?
When evaluated at a positive input, arcsin, arccos, and arctan return a reference angle rather than the full answer. Since every trig value in Quadrant I is positive, a positive input can only correspond to a first-quadrant, acute result, which is exactly what a reference angle is. Finding the actual angle then requires combining that reference angle with a separately known quadrant.