the Fourier sine and cosine transforms
The plain Fourier transform is built for the whole line, from -infinity to +infinity. But many problems live only on the half-line x >= 0 — a semi-infinite rod, the air above the ground, the region outside a wall — with a single boundary condition at x = 0. The Fourier sine and cosine transforms are the versions tailored for exactly this situation, and which one you pick is decided entirely by the boundary condition at the origin.
Each uses a real oscillation rather than the complex e^(i*xi*x). The sine transform is integral from 0 to infinity of f(x) sin(xi*x) dx, and the cosine transform is integral from 0 to infinity of f(x) cos(xi*x) dx (up to a normalizing constant). The magic is in their differentiation rules, which automatically swallow the boundary value. For the sine transform, the second derivative f''(x) transforms to -xi^2 (sine transform) plus a term proportional to f(0) — so it cleanly handles a Dirichlet condition where you are told u(0). For the cosine transform, f''(x) transforms to -xi^2 (cosine transform) minus a term proportional to f'(0) — so it cleanly handles a Neumann condition where you are told the slope u_x(0). In short: sine transform for prescribed value, cosine transform for prescribed flux.
You can see them as the continuous cousins of the Fourier sine and cosine series, and equivalently as the ordinary transform applied to the odd or even extension of f across x = 0 — odd extension forces a zero at the origin (Dirichlet), even extension forces zero slope there (Neumann). The practical payoff: applying the right transform to a heat or wave problem on the half-line turns the PDE into an ODE in xi with the boundary data already built in, so you never have to chase boundary terms by hand.
A semi-infinite rod x >= 0 starts cold, u(x,0) = 0, and its end is suddenly held at temperature u(0,t) = T_0 (a Dirichlet condition). The sine transform turns u_t = k u_xx into an ODE whose forcing carries the value T_0, and inverting gives the classic error-function profile u(x,t) = T_0 erfc(x/sqrt(4 k t)) — heat creeping in from the held end.
Sine transform for a prescribed value at x=0, cosine transform for a prescribed slope.
Pick the wrong one and the boundary term in the differentiation rule will not match your condition, leaving a stray f(0) or f'(0) you cannot evaluate. Match the transform to the condition: sine with Dirichlet, cosine with Neumann.