A planet is located 1 AU from its star. The star is located 20 light years from Earth. Determine the angle expressed in radians?

Angle of what, you idiot !

Each light year = 63240 AU.

The radius of that orbiting planet is (Radian measure is 'the' ratio)

= 1 AU / [20 x 63240] AU = 1/126480 Radian

= 1.63 Arc seconds, after converting radians to Arc seconds.

So if you have a telescope that can resolve this angle you can see them (planet & star) separately and I haven't consider a dozen other aspects that would obscure the view immeasurably.

For very small angles, astronomers use the shortcut that Sin(x) = x

(this is, technically, only true at exactly x=0, but in practice, once you are within 0.001 radians, it is close enough... depending what you are doing, of course)

We assume you want the maximum possible angle, when the planet is furthest from its star:

Sin(x) = opposite / hypotenuse

sin(x) = orbital radius / distance to star

sin(x) = 1 AU / (20 * 63241 AU) = 0.000000790626...

You could calculate the value of x (the angle in radians) by taking the arcsine of this value, but you will find a difference only at (and after) the 18th decimal.

0.00000079062633418194 = sin(x)

0.00000079062633418202 = x

That is why we don't bother, for such small angles. We go directly to the shortcut x = opp/hyp.

If you want to use the traditional units (degrees, minutes and seconds), then these small angles will be in seconds of arc, or even in "milliarcseconds" (1 mas = 1/1000 of a second), a unit still used in astronomy - specially in radioastronomy.

There are 206264.8" in one radian

x = 0.0000007906263... radians

x = 0.00000079062633418 * 206264.8" = 0.163078" (arc seconds)

x = 163.078 mas (milliarcseconds)

All these numbers mean the same angle.

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There are 63,241 astronomical units in one light-year

There are 206,264.8 astronomical units in one parsec, because the parsec is based on using an angle of 1" with the "opposite side" being 1 AU.

(3.26/20) arcseconds • 4.848[-6] arcseconds/radians = 7.90[-7] radians

zero.

(always questions from teachers are incomplete.)

I think it 4.23 radians bro. Don't quote me though.