document.write( "Question 1067578: Please help me with this word problem.\r
\n" ); document.write( "\n" ); document.write( "The index of refraction $\"+n+\"$ of a transparent material is the ratio of the speed of light in a vacuum to the speed of light in the material. For the triangular prism in the picture, n = 1.5 and a = 60°. Identify the angle θ for the glass prism if ( see image linked below ) thanks a bunch.\r
\n" ); document.write( "\n" ); document.write( "image: https://s28.postimg.org/q8pbxbknx/word_problem.png \n" ); document.write( "

Algebra.Com's Answer #683045 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
The picture:
\n" ); document.write( " $\"n=sin%28theta%2F2%2Balpha%2F2%29%2Fsin%28theta%2F2%29\"$

\n" ); document.write( " $\"alpha=60%5Eo\"$ = prism angle crossed by the light
\n" ); document.write( " $\"alpha%2F2=60%5Eo%2F2=30%5Eo\"$
\n" ); document.write( " $\"theta\"$ = angle of deviation of the light as it goes through the prism
\n" ); document.write( " $\"sin%28alpha%2F2%29=sin%2830%5Eo%29=1%2F2=0.5\"$
\n" ); document.write( " $\"cos%28alpha%2F2%29=cos%2830%5Eo%29=sqrt%283%29%2F2=0.5sqrt%283%29\"$ \r
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "To calculate $\"sin%28theta%2F2%2Balpha%2F2%29\"$ we need the trigonometric identity formula for a sum of angles:
\n" ); document.write( " $\"sin%28A%2BB%29=sin%28A%29%2Acos%28B%29%2Bcos%28A%29%2Asin%28B%29\"$ .
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\n" ); document.write( "So substituting the expression above,
\n" ); document.write( "and the value $\"1.5\"$ for $\"n\"$
\n" ); document.write( "in the formula for refractive index, we get
\n" ); document.write( "

.
\n" ); document.write( "Multiplying both sides of the equal sign times $\"2sin%28theta%2F2%29\"$ ,
\n" ); document.write( "we get the equivalent equation
\n" ); document.write( " $\"3sin%28theta%2F2%29=sqrt%283%29%2Asin%28theta%2F2%29%2Bcos%28theta%2F2%29\"$
\n" ); document.write( "Now, dividing both sides of the equation by $\"cos%28theta%2F2%29\"$ ,
\n" ); document.write( "we get the equivalent equation
\n" ); document.write( "

,
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"3tan%28theta%2F2%29=sqrt%283%29%2Atan%28theta%2F2%29%2B1\"$ ,
\n" ); document.write( "so
\n" ); document.write( " $\"3tan%28theta%2F2%29-sqrt%283%29%2Atan%28theta%2F2%29=1\"$ --> $\"%283-sqrt%283%29%29%2Atan%28theta%2F2%29=1\"$ --> $\"tan%28theta%2F2%29=1%2F%283-sqrt%283%29%29\"$
\n" ); document.write( "From
\n" ); document.write( " $\"tan%28theta%2F2%29=1%2F%283-sqrt%283%29%29\"$ , or the more elegantly written equivalent
\n" ); document.write( " $\"tan%28theta%2F2%29=%283%2Bsqrt%283%29%29%2F6\"$ ,
\n" ); document.write( "we can calculate an approximate value for $\"tan%28theta%2F2%29\"$ ,
\n" ); document.write( "for which your calculator would give you an approximate value for $\"theta%2F2\"$ ,
\n" ); document.write( "which can be used to calculate an approximate value for $\"theta\"$ .
\n" ); document.write( "The approximate values would be:
\n" ); document.write( " $\"tan%28theta%2F2%29=0.788675\"$ , $\"theta%2F2=38.26%5Eo\"$ ,
\n" ); document.write( "and $\"highlight%28theta=76.52%5Eo%29\"$ .
\n" ); document.write( "
\n" ); document.write( "An exact value would have to be expressed as an inverse tangent function
\n" ); document.write( " $\"theta=2%2Atan%5E%28-1%29%28%283%2Bsqrt%283%29%29%2F6%29\"$ ,
\n" ); document.write( "or if you use the double angle trigonometric identity to calculate that $\"tan%28theta%29=%2830%2B14sqrt%283%29%29%2F13\"$ ,
\n" ); document.write( " $\"theta=tan%5E%28-1%29%28%2830%2B14sqrt%283%29%29%2F13%29\"$ .
\n" ); document.write( "Hopefully no instructor would make you go through all that trouble. \n" ); document.write( "

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