The astrononomical parsec is derived from the following triangle:

The baseline is the Earth-Sun distance, or 1 AU. The triangle is a right triangle. The apex angle at right is 1 second (1") of arc. This is the definition of the parallax triangle. We will use the trigonometric tangent function to describe the triangle; divide the length of the au by the length of the parsec and get the tangent of 1" of arc. Write it as AU/parsec=tan(1"). Both values on the left side are distances.

Given that the tangent of an angle is opposite/adjacent and the opposite side is 1 AU, then the parsec is 1AU/tan(1"). This is rearranging the above equation and defines the parsec (pc).

Let's do this. 1AU/tan(1") is 1/4.8481368x10^-6, which yields 206264.8 AU/parsec.

But - how long is a parsec in more familiar distance units? Let's see if we can derive this in Light-Years (LY). It's going to take a number of units conversions (calculator button-pushing). Note: minutes will be used for distances covered at light-speed (c).

First - we must express the AU in terms of light-years. We'll use 150,000,000 km for the AU. The speed of light (c) is 300,000 km/sec. Therefore, dividing km by km/sec will yield seconds. so 150,000,000/300,000 yields 500 light-seconds for the AU. Divide by 60 to get 8.333... Light-Minutes for the AU. We'll call the AU 8.333... light-minutes (LM/AU).

Next - we need to find out how many AUs there are in a Light-Year. Since an AU is 8.333... LM/AU we now need to find the number of minutes in a year. Easy: 60*24*365.2422 gives 525,948.8 LM/LY. Let's divide that by 8.33333 LM/AU (525948.8/8.33333) and get 63,113.8 AU/LY. Look at the units here: We're dividing minutes/year by minutes/AU. This gives AU/LY.

We're almost there. Now divide AU/parsec by AU/LY to get LY/parsec. 206264.8/63118.8 yields 3.26788 LY/parsec. It's just calculator button-pushing, being sure to keep your units straight.