Visual Unit Circle Sine and Cosine Without Math Libraries

Trig from the unit circle — sine as y-coordinate, cosine as x-coordinate

One picture

Put a point on a circle of radius 1. Rotate it by angle θ.

cos(θ) is the point's x-coordinate (horizontal distance from center)
sin(θ) is the point's y-coordinate (vertical distance from center)

point = (cos θ, sin θ)

Simple idea

Cosine is sideways position. Sine is vertical position. That's the whole thing.

Key angles

Angle	cos	sin
0°	1	0
30°	√3/2 ≈ 0.866	0.5
45°	√2/2 ≈ 0.707	√2/2 ≈ 0.707
60°	0.5	√3/2 ≈ 0.866
90°	0	1
180°	-1	0
270°	0	-1
360°	1	0

Quadrant signs

Quadrant	x (cos)	y (sin)
I (0–90°)	+	+
II (90–180°)	–	+
III (180–270°)	–	–
IV (270–360°)	+	–

Memory: "All Students Take Calculus" — All / Sin / Tan / Cos are positive in quadrants I/II/III/IV respectively.

Lookup table implementation

import math  # only for building the table — remove for production

# Build a table at startup for 0–360 degrees
SIN_TABLE = [math.sin(math.radians(d)) for d in range(361)]
COS_TABLE = [math.cos(math.radians(d)) for d in range(361)]

def sin_lookup(degrees):
    d = int(degrees) % 360
    return SIN_TABLE[d]

def cos_lookup(degrees):
    d = int(degrees) % 360
    return COS_TABLE[d]

For truly no-library sine, use the Taylor series approximation:

PI = 3.141592653589793

def sin_approx(x):
    """Taylor series for sin(x), x in radians. Works well for small x."""
    x = x % (2 * PI)  # wrap to [0, 2π]
    return x - x**3/6 + x**5/120 - x**7/5040

def cos_approx(x):
    """Taylor series for cos(x), x in radians."""
    x = x % (2 * PI)
    return 1 - x**2/2 + x**4/24 - x**6/720

Using sine and cosine for rotation

To move a point in a circle (e.g., orbiting, rotating a sprite):

def point_on_circle(cx, cy, radius, angle_degrees):
    PI = 3.141592653589793
    theta = angle_degrees * PI / 180
    x = cx + radius * cos_approx(theta)
    y = cy + radius * sin_approx(theta)
    return x, y

Common mistakes

Confusing which is x and which is y. Cosine = x (horizontal), Sine = y (vertical).
Passing degrees to functions that expect radians (Python's math.sin expects radians).
Forgetting that both repeat with period 360° (or 2π radians) — values outside this range wrap around.

Results