An angle is the amount of rotation of a ray from its initial side to its terminal side. Drag the slider to rotate the terminal side and watch the degree and radian measures update together. Anticlockwise rotation is positive; clockwise is negative.
Notice: one full revolution = 360° = 2π radians. The point on the circle traces the same path whether you think in degrees or radians — they are just two different rulers for measuring the same rotation.
For a point P(a, b) on the unit circle with ∠AOP = x radians, we define cos x = a and sin x = b. Drag the slider (or the point itself) to explore how sin, cos, tan and the reciprocal functions change as x sweeps around the circle.
| Function | Value |
|---|---|
| sin x | - |
| cos x | - |
| tan x | - |
| cosec x | - |
| sec x | - |
| cot x | - |
cos²x + sin²x = 1 → check: -
| x | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
|---|---|---|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |
| tan | 0 | 1/√3 | 1 | √3 | ND | 0 | ND | 0 |
Move the slider to place the terminal side in each quadrant and see which functions are positive and which are negative. This is the famous "All Sin Tan Cos" (ASTC) rule read anticlockwise from quadrant I.
| I | II | III | IV | |
|---|---|---|---|---|
| sin x | + | + | – | – |
| cos x | + | – | – | + |
| tan x | + | – | + | – |
Select a function to see its graph, and watch a rotating radius on the unit circle (left) generate the curve (right) point by point. All trig functions are periodic: sin and cos repeat every 2π; tan and cot repeat every π.
Pick two angles x and y and verify the identities numerically and visually. These identities let us break any angle into a sum/difference of angles we already know.
sin 15° = sin(45° − 30°) = sin45°cos30° − cos45°sin30° = (√6 − √2)/4 ≈ -
These formulas come directly from the sum formulas by setting y = x (double angle) or using 3x = 2x + x (triple angle). Change x below and watch both sides of each identity match.
The curve shows y = sin x (cyan) versus y = sin 2x (yellow) — notice sin 2x completes two full cycles in the space sin x completes one, matching the "doubled" angle.
These convert a sum/difference of sines or cosines into a product (useful for solving equations), and vice versa. Adjust x and y and confirm both sides agree.