u/Independent_Ball7895

For the context : I'm currently doing 1st yr physics undergrad and an opportunity came around for me to also apply for aerospace course which is for the first 2 years conducted in online with physical lab sessions and if I qualify their exams with necessary credit points I'd be called for an interview to continue the full bs aerospace course.

I'd be lying if I said I'm not into aerospace but I'm more into its research area such as nuclear propulsion systems or turbulence.

I'm afraid I don't know how exactly to get to that position or if at all it's sustainable.

Suppose say, I go along with this course and complete the program, how would I have to proceed to those kinds of research?

I can see it could easily be proved with the cos(x+y) relation but I want to know it otherwise.

Here's how i imagined it:

Say, there's 2 axes, the horizontal axis is epsilon and the vertical axis eta. There's a line passing through the origin to the 2nd quadrant making an angle of x with the vertical axis and x+π/2 with the horizontal axis. Both these angles point to the same coordinates say (-epsilon,eta).

Therefore cos(x)=eta and sin(x)= -epsilon.

Now how do I further go on about it?

Or is there someother way to picture this?

I think the trigonometric ratios are deduced through the concept of similarity of a triangle but I do not understand how an angle gets related to those ratios in the Right triangle