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March 23rd, 2015, 01:35 AM  #1 
Newbie Joined: Mar 2015 From: Australia Posts: 5 Thanks: 0  Sine to find critical angle?
Good day. If light travels through an dense material lile silicon, to meet air, it may not leave the silicon but reach the critical angle and remain in the silicon, refracted at 90 degrees to the "normal" line which is at 90 degrees to the two mediums the light goes through. Like a pond air above, perfectly flat and straight up and down is the "normal". Going at angle a1, the light reaches the air, and refracts to 90 degrees to the normal. Completely within the silicon, not reaching the air. The "Critical Angle". Also angle or a2 is 90 degrees and sin90 = 1. arrow to sina1 = N1/N2 sina1 = N1/N2 N = the density of the mediums for the light. How can I find a1, the first angle? Last edited by Novatian; March 23rd, 2015 at 01:46 AM. 
March 23rd, 2015, 04:13 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,982 Thanks: 1575 
Are you familiar with the inverse sine function, $\arcsin(x)$ or $\sin^{1}(x)$ ? $\displaystyle a_1=\arcsin\left(\frac{n_1}{n_2}\right)$ The Critical Angle 
April 15th, 2015, 03:50 AM  #3 
Newbie Joined: Apr 2015 From: 6594 Ellies Way Fairfax Station, VA 22039 Posts: 5 Thanks: 2 
Hello Novatian The derivation is shown below. ni *â€¢ sine(Î˜i) = nr â€¢ sine (Î˜r) ni â€¢ sine(Î˜crit) = nr â€¢ sine(90 degrees) ni â€¢ sine(Î˜crit) = nr sine(Î˜crit) = nr/ni Î˜crit= sine1 (nr/ni) = invsine (nr/ni) I hope this will help you for your solution. 

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angle, critical, find, sine 
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