Final answer:
The question is focusing on the inclination of principal axes in the plane of a right triangle at the vertex with the right angle, calculated using the formula ½ tan⁻¹(ab/a²-b²).
Step-by-step explanation:
The student is asking about the inclination of the principal axes in the plane of a right triangle, specifically in relation to the sides at the vertex where the right angle is located. The entity 'principal axis' typically refers to the major axes of an ellipse or similar conic section, but in the context of this triangle, it seems to imply the axes that would be rotated relative to the sides of the triangle.
Using the formula given for the angle of inclination, ½ tan⁻¹(ab/a²-b²), we can derive the inclination of the principal axis by understanding that this angle is derived from trigonometric considerations and properties of right triangles.
The Pythagorean theorem, a² + b² = c², is fundamental in this discussion, as it relates the sides of the right triangle, and understanding the geometry allows us to apply the trigonometric principles accordingly. The inverse tangent function (tan⁻¹) is used to calculate the angle from a given ratio of opposite to adjacent sides in a right-angled triangle.