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Some Applications Of Trigonometry

Question

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Evaluate:
$fraction numerator cos space 45 degree over denominator sec space 30 degree space plus space cosec space 30 degree end fraction.$

Solution

fraction numerator cos space 45 degree over denominator sec space 30 degree plus cosec space 30 degree end fraction

equals space fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator begin display style fraction numerator 2 over denominator square root of 3 end fraction end style plus begin display style 2 over 1 end style end fraction space equals space fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator begin display style fraction numerator 2 plus 2 square root of 3 over denominator square root of 3 end fraction end style end fraction

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equals space fraction numerator 2 square root of 6 minus 2 square root of 18 over denominator 8 minus 24 end fraction equals fraction numerator 2 square root of 6 minus 6 square root of 2 over denominator negative 16 end fraction equals space fraction numerator 2 left parenthesis square root of 6 minus 3 square root of 2 right parenthesis over denominator negative 16 end fraction equals fraction numerator 3 square root of 2 minus square root of 6 over denominator 8 end fraction.

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