How can I calculate $\alpha=\arccos\left(-\frac{1}{4}\right)$ without using a calculator? How can I calculate $\alpha$, without using a calculator?

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I know $x = -\frac{1}{4} \implies y= \frac{\sqrt{15}}{4}, $ now how can I calculate $$\arccos\left(-\frac{1}{4}\right) = \alpha,\quad \arcsin\left(\frac{\sqrt{15}}{4}\right) = (180° - \alpha),$$  without using a calculator? How did the Greeks to calculate the angle?

The equation of the unity circle is $$x^2+y^2=1$$
then you have a function $f(x)=\sqrt{1-x^2}$ defined on $[-1/4,1]$ the length of a curve by defintion is $$ \int_{-1/4}^1\sqrt{1+(f'(x))^2}dx$$ 
We have $f'(x)=\frac{-x}{\sqrt{1-x^2}}$, then $$\alpha=\int_{-1/4}^1\sqrt{1+\frac{x^2}{1-x^2}}dx$$
After simplification: $$\alpha=\int_{-1/4}^1\frac{1}{\sqrt{1-x^2}}dx=\arcsin(1)-\arcsin(-1/4)$$
Hence $$\alpha=\frac{\pi}{2}-\arcsin(-1/4)$$
The problematic consists in how to approximate $\arcsin(-1/4)$;
we can use a Taylor series: decomposition of $\arcsin(x)$, one has $$\arcsin(x)= x+ 1/2 (x^3/3) + (1/2)(3/4)(x^5/5) + (1/2)(3/4)(5/6)(x^7/7) ...$$
$$\arcsin(-1/4)= -1/4+ 1/2 ((-1/4)^3/3) + (1/2)(3/4)((-1/4)^5/5) + (1/2)(3/4)(5/6)((-1/4)^7/7)...=   -    0.252$$
Then $$\alpha=  1.822 rad$$
