So, a light year is #5.879*10^12# miles and an AU is #9.296*10^7# miles. Now in order to make our calculations a little easier, let's express the light year in terms of AUs so at least one side of the triangle we will be using has a side of length 1. That tells us that a light year is 63241.1 AUs, making the distance to Alpha Centauri, in AUs equal to:
#63241.1 * 4.365 = 276047.4015# AUs
Now we have a right triangle with one side starting at Alpha Centauri (A) and ending at the Sun (S), another side starting at the Sun and ending at the Earth (E), and the third side starting at the Earth and ending at Alpha Centauri. Let's make the angle #/_ASE# equal to #90^o#. Now we can calculate the third side of this triangle, or the hypotenuse, using Pythagoras' Theorem:
#A^2+B^2=C^2#
#C^2=276047.4015^2+1^2#
#C=sqrt(276047.4015^2+1^2)#
#C=276047.40150181128312485984269162#
Now, the angle we want is #/_SAE#, so, using the formula for the sine:
#sin(x)="opposite side"/"hypotenuse"#
#sin(x)=1/C=3.6225662497078006614860942772394*10^-6#
In order to convert a sine to the angle it represents we use the arcsine function (#sin^-1# on your calculator). With that we get that the angle x is equal to 0.00020755775711524564 degrees. This is not good for us since the answer we need is in arcseconds. To convert one to the other you need to know that 1 arcsecond is comprised of 0.00027777777777778 degrees. So we divide our angle by that number and we get:
0.74720792561487826563392497470966 or to approximate 0.75 arcseconds.
Some resources to make this easier are:
Arc Sine Calculator
Degree/Arc Second Converter
