My question stems from the following graphic:
It aims to explain the following relationship:
$$\hat r = cos\ \theta \ \hat i + sin\ \theta \ \hat j$$ $$\hat \theta = -sin\ \theta \ \hat i + cos\ \theta \ \hat j $$
Now, judging from the graphic, $\hat r$ is derived from considering trigonometric properties of the triangle formed by hypotenuse $\hat r$ and its sine and cosine components as its legs. Since $\theta$ is at the tail of $\hat r$, its $x$ component is $cos \ \theta$ and its y component $sin \ \theta$. However, going by the logic of the derivation of $\hat r$, I'd expect the $x$ component of $\hat \theta$ to be $cos\ \theta_2$ and $y$ to be $sin\ \theta_2$. Note that I also added the subscript $2$ to differentiate it between the $theta$ used in finding the components of $\hat r$, but it highlights my two main confusions.
- Why is theta taken to be at the head of $\hat \theta$ so that its $y$ and $x$ components are the way they are, and if that's not why $x$ is $sin\ \theta$ and $y$ is $cos\ \theta$, what's the reason?
- Sine and cosine $\theta$ were written twice in each triangle in question in this graphic. Are they the same magnitude? And if not, why haven't they been distinguished by something like $\theta$ and $\theta_2$?
I feel like its highlighting a key trigonometric property I've forgotten, but since I don't know exactly what I'm not understanding I can't look up such a detail.