# Calculate the angle between the lines

I need to calculate the angle x. I feel like I'm missing something basic since this is supposed to be an easy exercise but I can't seem to get it. It's under the chapter trigonometry.

AB is 8 units long, the radius is 5. I was looking for right angle triangles, but didn't find a way to make use of the info.

• Note that SA=SB=5, because that’s the radius Dec 11, 2023 at 14:30
• First find RB. Then look at triangle RSB. Dec 11, 2023 at 17:03

You're correct to look for right-angled triangles. In particular, as shown below in a modified version of your diagram, the midpoint of $$AB$$ is labeled as $$C$$, and with the line segments of $$SA$$, $$SC$$ and $$SB$$ added.
As Divide1918's comment indicates, $$\lvert SA\rvert = \lvert SB\rvert = 5$$ as they are the radii. Thus, $$\triangle ASB$$ is isosceles, so with $$C$$ being the midpoint of $$AB$$, then $$\measuredangle SCB = 90^{\circ}$$. Since $$\lvert CB\rvert = 4$$, then using the Pythagorean theorem with $$\triangle SBC$$, we get that
$$\lvert SC\rvert = \sqrt{5^2 - 4^2} = \sqrt{9} = 3$$
With $$\triangle SCR$$ being right-angled, we have
$$\sin(x) = \frac{\lvert SC\rvert}{\lvert SR\rvert} = \frac{3}{12} = 0.25 \;\;\;\to\;\;\; x = \sin^{-1}(0.25)$$