Question

Procedure 4. How small is small? (omit uncertainty, we're only making a point) This idea is so useful, it has a name, the Small Angle Approximation. What we want to do is approximate sin x by x, so we can get the trigonometry out of the calculation. Mathematically. we're going to take the first term in the Taylor series, but that's math for another time. If you're interested, there's great video on YouTube for using the Taylor series to approximate functions. Plot y = sin x and y = x on one graph, in Excel, (careful, Excel only understands radians). Ok, that might be asking a lot of inexperienced users. Do it this way. Fill one column with angles, x, from 0 to 90, in the next column fill with the formula Sin (x*PI(180) (converting the angle to radians), where x is in the cell before. You did this kind of thing in the first lab. Insert a scatter plot, right click and select 'select data' to enter 2 series. One will be x vs x and the other x vs sin x. You can scan through the columns and take a trial and error approach to answer the second part. If you still can't figure it out how to do this, ask in Blackboard. The plot is your data for the procedure. For analysis, give the angle for which sin x is 99% of x. In your conclusions, discuss how'small an angle should be to justify using the Small Angle Approximation

Answer 1

For, angle 0^o, the value of sin (x)  is 99% of x. The small angle approximation is approximately true in the limit where the angle approaches zero.

x in radian х Sin (x) 0 0 0 y = x and y = sinx y = x y = Sin (x) 30 0,523599 0,5 60 1,047198 0,866025 90 1,570796 1 120 2,094395 0,866025 150 2,617994 0,5 180 3,141593 1,23E-16 210 3,665191 -0,5 240 4,18879 -0,86603 270 4,712389 -1 300 5,235988 -0,86603 330 5,759587 -0,5 360 6,283185 -2,5E-16 7 6 5 4 3 2 1 0 0 50 100 150 200 250 300 350 -1 400 -2