त्रिकोणमिति का परिचय

Question

त्रिभुज ABC में, जिसका कोण B समकोण है, यदि $tanA space equals space fraction numerator 1 over denominator square root of 3 end fraction$ तो निम्नलिखित के मान ज्ञात कीजिए:
(i) sin A cos C + cos A sin C
(ii) cos A cos C - sin A sin C

Solution

हमें प्राप्त है:
$tanA space equals space fraction numerator 1 over denominator square root of 3 end fraction equals BC over AB$
मान लीजिए: $AB space equals space square root of 3 straight K comma space space space BC space equals space straight K$
पाइथागोरस प्रमेय के प्रयोग से:
$AC squared space equals space AB squared plus BC squared space space space space space space space space space equals space left parenthesis square root of 3 straight K right parenthesis squared plus left parenthesis straight K right parenthesis squared space space space space space space space space space equals space 3 straight K squared plus straight K squared space equals space 4 straight K squared$
AC = 2K

$space sin space straight A space equals space BC over AC equals fraction numerator straight K over denominator 2 straight K end fraction equals 1 half space sin space straight C space equals space AB over AC equals fraction numerator square root of 3 straight K over denominator 2 straight K end fraction equals fraction numerator square root of 3 over denominator 2 end fraction अब comma space cos space straight A space equals space AB over AC space equals space fraction numerator square root of 3 straight K over denominator 2 straight K end fraction equals fraction numerator square root of 3 over denominator 2 end fraction$
और, $cos space straight C space equals space BC over AC equals fraction numerator straight K over denominator 2 straight K end fraction equals 1 half$
अब,
(i) sin A cos C + cos A sin C
$equals space 1 half cross times 1 half plus fraction numerator square root of 3 over denominator 2 end fraction cross times fraction numerator square root of 3 over denominator 2 end fraction equals space 1 fourth plus 3 over 4 space equals space 4 over 4 space equals space 1$
(ii) cos A. cos C - sin A. sin C
$equals space fraction numerator square root of 3 over denominator 2 end fraction cross times 1 half minus 1 half cross times fraction numerator square root of 3 over denominator 2 end fraction equals space fraction numerator square root of 3 over denominator 4 end fraction minus fraction numerator square root of 3 over denominator 4 end fraction space equals space 0.$

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