Trigonometry

An angle in radians is arc length divided by radius: θ = s/r. On the unit circle the radius is 1, so the angle equals the signed arc length. A full turn travels 2π radii: 360° = 2π radians, and a half turn is 180° = π radians. Positive angles turn counterclockwise; negative angles turn clockwise.

Convert degrees to radians by multiplying by π/180; convert radians to degrees by multiplying by 180/π. NumPy’s trigonometric functions expect radians. Radians also make the calculus identities natural: the derivative of sin θ is cos θ without an extra conversion factor.

The point at angle θ on the unit circle is (cos θ, sin θ). Cosine is the horizontal component; sine is the vertical component. Their squares add to 1 by the Pythagorean theorem. Both repeat after 2π; cosine leads sine by π/2. The linked graph markers show the same point’s coordinates plotted against angle.

tan θ = sin θ / cos θ. It is the slope of the line through the origin and the unit-circle point. That line meets the vertical tangent x = 1 at height tan θ. When cos θ = 0, the ratio is undefined: θ = π/2 + kπ. The graph has separate branches and vertical asymptotes, not connecting lines through infinity. Tangent repeats after π. The diagram clips values outside its visible bounds.

Here u lies along the positive x axis and v makes angle θ with it. The signed projection of v onto u’s direction is |v| cos θ. Multiplying by |u| gives u · v = |u| |v| cos θ. In coordinates this is u_x v_x + u_y v_y. Aligned vectors have a positive dot product, perpendicular vectors have zero, and opposite vectors have a negative dot product. The orange segment is the projection, offset slightly from the axis for visibility.

Cosine similarity divides the dot product by both vector lengths. It is 1 for the same direction, 0 for orthogonal vectors, and -1 for opposite directions. Changing a nonzero vector’s length changes the dot product but not cosine similarity. A zero vector has no direction, so its cosine similarity is undefined. In embedding spaces the same formula works in many dimensions; whether angular similarity captures meaning depends on the representation and task.

The parallelogram spanned by u and v has area |u| |v| |sin θ|. Its signed 2D determinant is u_x v_y - u_y v_x. When the vectors are embedded in the xy plane in 3D, this is the z component of u × v: positive points out of the screen, negative into it. The full cross product is a vector, not a similarity score. Its magnitude vanishes for parallel and antiparallel vectors and is largest for perpendicular vectors of fixed length.

Continue to Calculus for derivatives and Linear Algebra for vector and matrix computations.