The unit circle: sin, cos, tan at standard angles
One circle of radius 1 pins down every trig value you'll meet at school or in first-year university.
What this shows
The unit circle is the circle of radius 1 centred at the origin of the (x, y) plane. For any angle θ measured counter-clockwise from the positive x-axis, the point where the angle's terminal ray meets the circle has coordinates (cos θ, sin θ). Tangent is just sin θ divided by cos θ, which geometrically is the slope of the terminal ray.
At the five standard angles — 0°, 30°, 45°, 60°, 90° — the coordinates come out to exact algebraic values:
0° → (1, 0) → cos 0 = 1, sin 0 = 0 30° → (√3/2, 1/2) → cos 30 = √3/2, sin 30 = 1/2 45° → (√2/2, √2/2) → cos 45 = √2/2, sin 45 = √2/2 60° → (1/2, √3/2) → cos 60 = 1/2, sin 60 = √3/2 90° → (0, 1) → cos 90 = 0, sin 90 = 1
These five values come up so often that memorising them — or, better, being able to re-derive them from the 30-60-90 and 45-45-90 right triangles — saves enormous amounts of time.
Where it shows up
Anywhere an angle isn't part of a right triangle, the unit circle is the natural home for it: signed values for angles above 90°, periodic identities (sin(θ + 360°) = sin θ), the sign pattern of each function across the four quadrants, and the conversion between radians and degrees.
In physics and engineering the unit circle is the spine of everything that oscillates — alternating current, simple harmonic motion, Fourier decompositions — because rotating a point at constant angular speed around the unit circle projects onto the x-axis as a perfect cosine wave.
Frequently asked questions
Why radius 1?
Because then sin and cos are just the y and x coordinates of the point, with nothing to scale. Any other radius works too — you just have to divide out the radius to get the trig values back.
What about angles past 90°?
The same rule applies: travel θ degrees counter-clockwise from the positive x-axis, read the coordinates of the meeting point with the circle. At 180° you reach (-1, 0); at 270°, (0, -1); at 360°, back to (1, 0). Negative angles go clockwise.
Do I have to memorise the values?
Not strictly — they all come from the 30-60-90 and 45-45-90 right triangles, which you can re-derive in a minute. But the first time you sit down to learn them, drilling them is faster than re-deriving every time.
Where does tan fit in?
tan θ = sin θ / cos θ, which geometrically is the slope of the terminal ray. So tan is undefined whenever the ray is vertical — at 90° and 270°. Some textbooks draw a separate vertical tangent line to the circle at (1, 0) to read tan directly off.