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PHY 202
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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General Question on Sines
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The textbook mentions numerous times that when the angle in question is small, the sine of that angle can be neglected and you can use the angle itself in its place. taking the sine or not.
For example, Ch 24 #3:
d*sin(theta) = m*lambda
(4.8*10^-5m)sin(0.013) = (1)*lambda
I initially calculated 1.09*10^-8m for lambda, but the textbook solution was 6.24*10^-7m, which occurs when the equation is (4.8*10^-5m)*(0.013) = (1)*lambda (ie: without taking the sine of 0.013).
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I take this to mean that the answer therefore should be very similar regardless of whether you use sine or not... However, when doing the textbook problems I found I was getting the wrong answers because I was consistently taking the sine when I was not supposed to. I'm just a little frustrated with this because I like following equations the same way each time. It seems like a gray area to me to judge whether an angle is small enough to neglect. When do we take the sine and when do we not? In the above example from Ch 24, the sine of 0.013 is 2.27*10^-4, which is different enough from 0.013 that I would never think to neglect.
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You appear to be taking the sine of the angle, with the angle in degrees rather than radians.
The sine of 0.013 is very close to 0.013. The sine of 0.013 degrees is not.
The key is that 0.013 does not mean 0.013 degrees. The default unit for measuring angles is the radian. Any angle which does not specify a unit is considered to be in radians.
If you put your calculator into radian mode you will see how this works out.
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A radian is the central angle for which the corresponding arc of a circle is equal to its radius.
An equilateral triangle has 60 degree angles at its vertices. If you curve out one of those sides to make an arc of the circle (the circle being centered at the opposite vertex) it will cause the other two sides to get a little closer to one another, thereby reducing the angle at that opposite vertex from 60 degrees to about 57.3 degrees.
We get the 57.3 degrees because the circumference of a circle is 2 pi times its radius. Thus there are 2 pi arcs around the circle each equal to the radius. Each therefore corresponds to 360 / 2 pi degrees, or 180 / pi degrees. If you divide this out you get the 57.3 degrees.
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So for example the angle 0.013, which means 0.013 radians, is approximately 0.013 * 60 degrees = 0.78 degrees.
More accurately it could be calculated at 0.013 * 57.3 degrees.
A completely accurate expression of the angle in degrees would be 0.013 radians * 180 deg / pi radians = .234 / pi degrees.
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Alternatively an angle of, say, 14 degrees would be about 14 deg / (60 deg / radian) = .23 radian. More accurately 14 deg / (57.3 deg/radian) = .244 rad.
The sine of this angle is .241.
So for this angle the sine differs from the angle by only a little over 1%.
Suggestion:
Check this out for angles of .1, .2, .3, .4, .5 and .6, corresponding to angles of about 6, 12, 17, 23, 29 and 34 degrees.
This will give you a pretty good baseline understanding the accuracy of the approximatoin
sin(theta) = tan(theta) = theta
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The analysis of the circular lens problem pretty much requires that we use this approximation to understand the analysis of a ray near the axis.
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