How can I solve $\langle \sin(pt),\cos(pt)\rangle = \langle \cos(qt),\sin(qt)\rangle$ How can I solve this vector equality where $p$ and $q$ are integers and $0\leq t\leq 2\pi$? $\langle \sin(pt),\cos(pt)\rangle = \langle \cos(qt),\sin(qt)\rangle$ I am trying to figure out how many times in a period they intersect.
Solution is now edited to contain both versions of the question you've asked

Version 1: $⟨sin(pt),cos(qt)⟩=⟨cos(pt),sin(qt)⟩$

You need the $x$ and $y$ components to be equal so you are solving the simultaneous equations:

$\sin(pt)=\cos(pt)$ and $\cos(qt)=\sin(qt)$

These become: $\tan(pt)=1$ and $\tan(qt)=1$

So $pt=\frac{pi}{4}+\pi n_1,n_1\in\mathbb{N}$ and $qt=\frac{\pi}{4}+\pi n_2,n_2\in\mathbb{N}$

So $t=\frac{pi+4\pi n_1}{4p}=\frac{pi+4\pi n_2}{4q}$

This will only have solutions for some values of $p$ and $q$. Cross multiplying and dividing by $\pi$ gives:

$q+4n_1=p+4n_2$

As $p$ and $q$ are integers this will have no solutions unless $p-q=4m,m\in\mathbb{Z}$.

Version 2: $⟨sin(pt),cos(pt)⟩=⟨cos(qt),sin(qt)⟩$

Here each vector represents a rotation by $pt$ and $qt$ respectively. As $p$ and $q$ are integers they will obviously be equal when $t=2\pi n,n\in\mathbb{Z}$. They will also be equal before then if $p$ and $q$ have a common factor. So the solution is:

If $p=q$ then $t\in\mathbb{R}$

If $p\neq q$ then $t=\frac{2\pi}{\gcd(p,q)}$
Solution is now edited to contain both versions of the question you've asked

Version 1: $⟨sin(pt),cos(qt)⟩=⟨cos(pt),sin(qt)⟩$

You need the $x$ and $y$ components to be equal so you are solving the simultaneous equations:

$\sin(pt)=\cos(pt)$ and $\cos(qt)=\sin(qt)$

These become: $\tan(pt)=1$ and $\tan(qt)=1$

So $pt=\frac{pi}{4}+\pi n_1,n_1\in\mathbb{N}$ and $qt=\frac{\pi}{4}+\pi n_2,n_2\in\mathbb{N}$

So $t=\frac{pi+4\pi n_1}{4p}=\frac{pi+4\pi n_2}{4q}$

This will only have solutions for some values of $p$ and $q$. Cross multiplying and dividing by $\pi$ gives:

$q+4n_1=p+4n_2$

As $p$ and $q$ are integers this will have no solutions unless $p-q=4m,m\in\mathbb{Z}$.

Version 2: $⟨sin(pt),cos(pt)⟩=⟨cos(qt),sin(qt)⟩$

Here each vector represents a rotation by $pt$ and $qt$ respectively. As $p$ and $q$ are integers they will obviously be equal when $t=2\pi n,n\in\mathbb{Z}$. They will also be equal before then if $p$ and $q$ have a common factor. So the solution is:

If $p=q$ then $t\in\mathbb{R}$

If $p\neq q$ then $t=\frac{2\pi}{\gcd(p,q)}$