Can someone explain why F=mg sin theta ? It is only simple trigonometry although it is a mechanics example.
Draw the right triangle that has mg as the hypotenuse and the projection of mg to the inclined plane as one of its legs. This projection (i.e., its absolute value) equals F. To find one of the acute angles in this triangle use the fact that two angles are equal if their sides are perpendicular. That is, if one angle consists of rays p and q, another angle consists of rays p' and q', p is perpendicular to p' and q is perpendicular to q', then the angle measures are equal.
No, I mean the projection of $\displaystyle m\vec{g}$ on the inclined plane equals (in absolute value and is opposite in direction to) F. The projection of $\displaystyle m\vec{g}$ on the line perpendicular to the plane equals R.
Look, mathematics is simpler than some philosophy or art critique because mathematicians use a single meaning of each word or expression they use. They don't chase after creating a complex bouquet of emotions with the help of finely crafted phrases full of word play and figures of speech. Such phrases are complicated because there are many ways of interpreting them and because they have to be taken as a whole. In contrast, in mathematics, if you understand the meaning of each individual term in a sentence, it should be easy to understand the whole sentence.
What is my point? It is useless to say, "I don't understand your explanation." The only way not to understand it is not to know a particular term. For example, if you don't know how to take a projection of a vector on a line, then, yes, you won't understand the explanation as a whole. But you should be able to point to one or several words and say, "I don't understand what that phrase means." Or if you don't understand how one claim follows from another, you can ask that specific question. Such questions are legitimate (but be sure you mastered the definitions of this subject matter). Saying "I don't understand your text in its entirety" is not legitimate.
So, please explain your difficulty more precisely.