# Using reference angles to find $\alpha$

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?

$$\sin(\beta)= \dfrac{3.77}{4}$$ and $$\tan (\theta) = \dfrac{1.34}{3.77} \implies \theta = 0.341$$ rad
• @Andrei $\beta$ is different from $\theta$ – ab123 Dec 7 '19 at 16:26
• @Lex_i $\sin$ and $\tan$ take positive values for angles in the range $(0, 90^\circ)$ – ab123 Dec 7 '19 at 16:27