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Derive the values of :-
1. sin18degree
2. cos18degree

Asked by Sukrit Saha 27th June 2016, 9:00 PM
Answered by Expert
Answer:
begin mathsize 12px style Let space straight x space equals space 18 degree
rightwards double arrow 5 straight x space equals space 90 degree
rightwards double arrow 2 straight x space plus 3 straight x space equals space 90 degree
rightwards double arrow 2 straight x space equals space 90 degree space minus space 3 straight x space
rightwards double arrow sin 2 straight x space equals space sin left parenthesis 90 degree space minus space 3 straight x right parenthesis
rightwards double arrow sin 2 straight x space equals space cos 3 straight x
rightwards double arrow 2 sinxcosx space equals space 4 space cos cubed straight x space minus space 3 cosx
rightwards double arrow 2 sinxcosx space minus space 4 space cos cubed straight x space plus space 3 cosx space equals space 0
rightwards double arrow cosx space left parenthesis 2 sinx space minus space 4 space cos squared straight x space plus space 3 right parenthesis space equals space 0
Since space cos x equals cos 18 degree not equal to 0 comma space we space get
rightwards double arrow 2 sinx space minus space 4 space left parenthesis 1 space minus space sin squared straight x right parenthesis space plus space 3 equals space 0
rightwards double arrow 4 space sin squared straight x space plus space 2 sinx space minus space 1 space equals space 0
This space is space straight a space quadratic space equation space in space sinx.
straight a space equals space 4 comma space straight b space equals space 2 comma space straight c space equals space minus 1
rightwards double arrow sinx space equals space fraction numerator negative 2 plus-or-minus square root of 4 plus 16 end root over denominator 8 end fraction equals space fraction numerator negative 2 plus-or-minus 2 square root of 5 over denominator 8 end fraction equals space fraction numerator negative 1 plus-or-minus square root of 5 over denominator 4 end fraction
Now space since space sin 18 degree space greater than space 0 space comma space as space 18 degree space lies space in space the space first space quadrant comma
rightwards double arrow sin 18 degree space equals space fraction numerator negative 1 plus square root of 5 over denominator 4 end fraction equals space fraction numerator square root of bold 5 bold minus bold 1 over denominator bold 4 end fraction
Now comma space sin squared 18 degree space plus space cos squared 18 degree space equals space 1
rightwards double arrow cos squared 18 degree space equals space 1 space minus space sin squared 18 degree
rightwards double arrow cos 18 degree space equals space square root of 1 space minus space sin squared 18 degree end root
rightwards double arrow cos 18 degree space equals space space square root of 1 space minus open parentheses fraction numerator square root of bold 5 bold minus bold 1 over denominator bold 4 end fraction close parentheses squared end root
rightwards double arrow cos 18 degree space equals space space square root of 1 space minus fraction numerator 5 space minus space 2 square root of bold 5 plus bold 1 over denominator 16 end fraction end root
rightwards double arrow cos 18 degree space equals space space square root of fraction numerator 16 space minus space 5 space plus space 2 square root of bold 5 minus bold 1 over denominator 16 end fraction end root
rightwards double arrow cos 18 degree space equals space space square root of fraction numerator 10 space plus space 2 square root of bold 5 over denominator 16 end fraction end root space equals fraction numerator square root of 10 space plus space 2 square root of bold 5 end root over denominator 4 end fraction end style
Answered by Expert 27th June 2016, 10:06 PM
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