Photoacoustic tomography (PAT) is based on the generation of ultrasound waves by heating an object with short light pulses. A three-dimensional image of the distribution of absorbed energy within the object is reconstructed from signals measured around the object with either point-like or extended, linear sensors. Limited angle artefacts arise when the curve or surface connecting neighbouring detectors is not closed around the object. For this case, there exists a 'detection region' in which all boundaries of an object are visible in the reconstruction. All straight lines passing through each point in this region intersect the detection curve or surface at least once. Although for these points an accurate reconstruction is possible, direct back projection leads to artefacts when some of the straight lines intersect the detection surface twice and others just once. In this work, special weight functions for direct, non-iterative back projection are presented that reduce these kinds of artefacts. A clear improvement in image quality is shown in simulations for three-dimensional (3D) imaging with point detectors and for two-dimensional (2D) imaging using line detectors compared to reconstruction without weight factors. For the 2D case also an experiment is shown. The presented weight factors make commonly used back projection formulae suitable for a more accurate reconstruction of the initial pressure distribution in cases where the detection aperture only covers a limited angle, and the region of interest lies within the detection region.