Sinusoidal and trapezoidal control references the method of commutating the BLDC motor. The BLDC motor can be pictured as a 3 cylinder engine with a hard crankshaft between them. To make the Engine move you fire your spark plug at the top of the cylinder to ignite the fuel and push the engine down and rotate the shaft. The timing between each cylinder is 120 deg apart because the cylinders are hard shafted. Now imagine that the BLDC motor is that Engine. the Use of hall effects tell you when to turn the switch and release voltage and current into the winding so motion occurs due to the magnetic field. This motion is abrupt and uncontrolled as the winding will pull current as fast as it can. It basically slams the magnetic field. Now imagine that instead of a switch we can throttle, "use an electrical valve", the method of voltage and current coming into the winding based on the position of the rotor and shaft. We throttle that power during the initial state where it is extremely loss related and match it to the BEMF of the motor making it highly efficient. It then forms a sinusoidal output in phase with the BEMF vs a trapezoidal output in phase with the BEMF. It requires an encoder feedback to do it but the control is much more efficient.
The trap commutated (by Halls) BLDC motor has up to 17% higher current ripple vs. sine commutated method due lower sensor resolution. But sine commutated (by encoder) motor need initialization procedure with each power up due to incremental as rule encoder. Such routine has some limitation for Z operated axis or applications with high load torques.