Tangent Checkers!

If you multiply a 45-degree angle by consecutive odd numbers, a checkered pattern can be made with the tangent! Check out the table below! Notice the pattern in the right column?

Angle [Number of Degrees] ÷ 45 Tangent of Angle
45°11
135°3-1
225°51
315°7-1
405°91
495°11-1
585°131
675°15-1
765°171
855°19-1
945°211
1035°23-1
1125°251

The tangents of these angles created a checkered pattern that goes: "+1, -1 , +1, -1, +1, -1..."! Every 2nd odd number that multiplies the 45-degree angle gives a tangent of -1. (Note: If you see no plus or minus to the left a number, then it's positive by default.)

And guess what: this math trick works in the negative direction of the number line!

Angle [Number of Degrees] ÷ 45 Tangent of Angle
-45°-1-1
-135°-31
-225°-5-1
-315°-71
-405°-9-1
-495°-111

What's the difference in this 2nd table? The pattern starts with -1 on the top instead of +1. Simple as that! It depends on where you start on the number line which one goes 1st on top in the table. But remember to skip every 90 degrees in the left column, because the common difference of the angle series is 90 degrees.

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© Derek Cumberbatch