I know how to find reference angles, but I'm having a hard time using reference angles to find an angle measure of interest.
For example, say we have an initial ray along the x-axis inside a unit circle and it meets the end of the circle at $P_1(4, 0)$, with its vertex of course at $(0, 0)$. We have the terminal ray passing counter clockwise at $P_2(-3.77, 1.34)$ on the circle. The angle measure here is $\alpha$. This is what we're looking for (rounded to one decimal).
I understand intuitively that $\alpha=\pi-\theta$, where $\theta$ is the angle measure of a constructed right triangle with height $1.34$ units, length $-3.77$ units, and hypotenuse $4$ units. But there is no way for me to complete this without already knowing what $\theta$ is. So it looks like I need to use a reference angle to find $\theta$, then use that to find $\alpha$.
First I tried using interior angles. One angle must be $90^\circ$, and so $\theta$ and its interior angle opposite to it (say $\beta$) must both add up to $90^\circ$. \begin{align*} \sin(\beta)=& -\frac{3.77}{4} \\ \arcsin(-\frac{3.77}{4})=& \beta \\ \beta =& -1.23 \ \text{or} \ 70.48^\circ \\ 70.48^\circ+\theta=&90^\circ \\ \theta =& 2.8 \ \text{rad} \\ \alpha =& \pi - 2.8 = 0.3 \ \text{rad} \end{align*} then I tried using an inverse function to find $\theta$ directly. \begin{align*} \tan(\theta)=&1.34/3.77 \\ \theta =& \arctan(-1.34/3.77) \\ \theta =&-0.341 \ \text{rad} \\ \alpha =& \pi-(-0.341) \\ \alpha =& 2.8 \ \text{rad}& \end{align*}
It seems 2.8 is the best answer I've gotten of all my attempts. Can anybody verify?
