I have three unit vectors in a problem:
$\hat{t}= (\cos(t),0,\sin(t)),$
$ \hat{m}= (0,0,1),$
$\hat{n}= (\sin(th),0,-\cos(th)).$
I know the solution for the problem is:
$(-\sin(2t)+ 5 \sin(2t-4th)+2 \sin(2 th))$
I want to write the answer in a form in which only used variables are unit vectors instead of anges. Could anyone help me?
Your solution can be expanded with the trig identities
$$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$$ $$\cos(2\theta) = \cos^2(\theta)-\sin^2(\theta)$$ $$\sin(\theta+\phi) = \sin(\theta)\cos(\phi)+\cos(\theta)\sin(\phi)$$ $$\cos(\theta+\phi) = \cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)$$
to be expressed in terms of $\sin(t)$, $\cos(t)$, $\sin(th)$, and $\cos(th)$.
We also have
$$\hat t\cdot\hat m = \sin(t)$$ $$\hat n\cdot\hat m = -\cos(th)$$ $$\hat t\cdot\hat n = \sin(th-t)$$
but this doesn't give us a way to express $\cos(t)$ or $\sin(th)$ with dot products. (We know that $\cos(t) = \pm\sqrt{1-\sin^2(t)}$ $= \pm\sqrt{1-(\hat t\cdot\hat m)^2}$, but cosine's sign is not determined by the sine.) So I think you must use the components
$$t_x = \hat t\cdot(1,0,0) = \cos(t)$$ $$n_x = \hat n\cdot(1,0,0) = \sin(th)$$