How to find line segment length $BC$ in angle $BAC$?

Question

So I'm getting frustrated with this problem I'm having with this geometry problem.

I have an angle $BAC$, and line segments $AB$ and $AC$ are equal to $1$ unit long. Given the angle $BAC$, what is the length of the line segment $BC$?

Answer 1

Draw a line from $A$ to the midpoint $M$ of $BC$. This gives you a right-angled triangle $ABM$. Letting $\theta = \angle BAC$ and $x=BC$, it follows that $$\sin\frac\theta2 = \frac x2,$$ so $x = 2\sin\frac\theta2$.

You can verify in your example that $2\sin\frac{315^\circ}2 = 0.765\ldots$