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Now that we have two functions that are well known and easy to work with, we know follow the curve with the circle with a constructed instrument or mathematically. Notice that as the circle rotates that the sine and cosine stay within the vertical center line. (This should be explored more!) So it is possible to construct an instrument to make it easier to draw this curve.

Now that the sine curve is drawn across the curves of the unknown object, it is time to compare the curve formed by the circle to the original sine curve.

Take the y values of the curved object and subtract it from the value of the sine curve we drew. Then take this newly modified curve of the sine and compare it to the actual sine curve itself. (For ease of use you can just run your circular instrument we created across a horizontal line.)

Add one to the value of the true sine curve to eliminate negative numbers. Subtract the only positive number sin curve from the curve we wrapped around the unknown curve object. Restated, subtract the sine curve from the curve formed by the rotating circle.

Repeat the steps for the cosine. And once the sine and cosine is found you can use the formula sin/cos = tan to find the tangent.

 

References: pic 1 and 2 from: "Rinehart Mathematical Tables, Formulas and Curves", Larsen, Rinehart 1955

pic 6 from: "Analytic Trigonometry with Applications Sixth Edition" Barnett, PWS Publishing Company 1995

 

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