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∆ABCમાં m∠C = 90 અને ten A = fraction numerator bold 1 over denominator square root of bold 3 end fraction, તો sin A અને cos B શોધો. 

∆ABCમાં m∠C = 90 લઈએ. 



અહીં, ten A = fraction numerator bold 1 over denominator square root of bold 3 end fraction

bold ten bold space bold A bold space bold equals bold space bold BC over bold AC bold space bold equals bold space fraction numerator bold 1 over denominator square root of bold 3 end fraction

bold BC over bold 1 bold space bold equals bold space fraction numerator bold AC over denominator square root of bold 3 end fraction bold space bold equals bold space bold k bold space bold left parenthesis bold ધ ા ર ો bold space bold ક ે bold right parenthesis bold space bold જ ્ ય ાં bold space bold comma bold space bold k bold space bold greater than bold space bold 0 bold space

bold AC bold space bold equals bold space square root of bold 3 bold space bold k bold space bold અન ે bold space bold BC bold space bold equals bold space bold k bold space

હવે પાયથાગોરસના પ્રમેય પરથી, 

AB2 = AC2 + BC2 

AB2 = (square root of bold 3k)2 + k

       = 3k2 + k2 

       = 4k2 

AB = 2k (k > 0) 

આમ, AC square root of bold 3k, BC = k અને AB = 2k 

sin A = bold BC over bold AB bold space bold equals bold space fraction numerator bold k over denominator bold 2 bold k end fraction bold equals bold space bold 1 over bold 2 bold space bold અન ે bold space bold cos bold space bold B bold space bold equals bold space bold BC over bold AB bold space bold equals bold space bold 1 over bold 2

જો cot θ = bold a over bold b, તો fraction numerator bold cos bold space bold theta bold space bold minus bold space bold sin bold space bold theta over denominator bold cos bold space bold theta bold space bold minus bold space bold sin bold space bold theta end fraction નું મૂલ્ય શોધો. 

cot θ = bold a over bold b આપેલ છે. 

fraction numerator bold cos bold space bold theta bold space bold minus bold space bold sin bold space bold theta bold space over denominator bold cos bold space bold theta bold space bold plus bold space bold sin bold space bold theta end fraction bold space bold equals bold space fraction numerator begin display style fraction numerator bold c bold o bold s bold space bold theta over denominator bold s bold i bold n bold space bold theta end fraction end style bold space bold minus bold space bold 1 over denominator begin display style fraction numerator bold c bold o bold s bold space bold theta over denominator bold s bold i bold n bold space bold theta end fraction end style bold plus bold space bold 1 end fraction bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold space bold because bold અ ં શ bold space bold અન ે bold space bold છ ે દન ે bold space bold italic s bold italic i bold italic n bold space bold italic theta bold space bold # bold space bold 0 bold space bold વડ ે bold space bold ભ ા ગત ાંં bold space bold right parenthesis

bold equals bold space fraction numerator bold c bold o bold t bold space bold theta bold space bold minus bold space bold 1 over denominator bold c bold o bold t bold space bold theta bold space bold plus bold space bold 1 end fraction

bold equals fraction numerator begin display style bold a over bold b end style bold minus bold 1 over denominator begin display style bold a over bold b end style bold space bold plus bold space bold 1 end fraction bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold left parenthesis bold because bold space bold italic c bold italic o bold italic t bold space bold italic theta bold space bold equals bold space bold a over bold b bold space bold મ ૂ કત ાં bold right parenthesis

bold equals bold space fraction numerator bold a bold space bold minus bold space bold b over denominator bold a bold space bold plus bold space bold b end fraction

જો cosec A = square root of bold 10 હોય, તો બાકીના પાંચ ત્રિકોણમિતીય ગુણોત્તર શોધો. 

∆ABC માં m∠B = 90 લઈએ. 

cosec A = bold AC over bold AB bold space bold equals bold space fraction numerator square root of bold 10 over denominator bold 1 end fraction



bold therefore fraction numerator bold AC over denominator square root of bold 10 end fraction bold space bold equals bold space bold BC over bold 1 bold space bold equals bold space bold k bold space bold left parenthesis bold ધ ા ર ો bold space bold ક ે bold right parenthesis bold space bold જ ્ ય ાં bold space bold k bold space bold greater than bold space bold 0
bold therefore bold space bold AC bold space bold equals bold space square root of bold 10 bold space bold k bold. bold space bold BC bold space bold equals bold space bold k bold space

હવે, AC2 = AB2 + BC2 

bold therefore bold space bold left parenthesis square root of bold 10 bold space bold k bold right parenthesis to the power of bold 2 = AB2 + k2 

AB = 3k (∵ k>0) 

આમ, AC = , BC = square root of 10k અને AB = 3k 

હવે, sin A • cosec A = 1 

bold italic s bold italic i bold italic n bold space bold italic A bold space bold equals bold space fraction numerator bold 1 over denominator bold cosec bold space bold A end fraction bold space bold equals bold space fraction numerator bold 1 over denominator square root of bold 10 end fraction

bold italic c bold italic o bold italic s bold space bold italic A bold space bold equals bold space fraction numerator bold A bold B over denominator bold A bold C end fraction bold space bold equals bold space fraction numerator bold 3 bold k over denominator square root of bold 10 bold k end fraction bold space bold equals bold space fraction numerator bold 3 over denominator square root of bold 10 end fraction bold space bold space

bold italic t bold italic e bold italic n bold space bold italic A bold space bold equals bold space fraction numerator bold B bold C over denominator bold A bold B end fraction bold space bold equals bold space fraction numerator bold k bold space over denominator bold 3 bold k end fraction bold space bold equals bold space bold 1 over bold 3

bold italic s bold italic e bold italic c bold space bold italic A bold space bold equals bold space fraction numerator bold 1 over denominator bold c bold o bold s bold space bold A end fraction bold space bold equals bold space fraction numerator square root of bold 10 over denominator bold 3 end fraction bold space bold અન ે bold space bold italic c bold italic o bold italic t bold space bold italic A bold space bold equals bold space fraction numerator bold 1 over denominator bold t bold e bold n bold space bold A bold space end fraction bold space bold equals bold space bold 3

∆ABCમાં AC = 5, BC = 13, m∠A = 90, તો ∠Bના તમામ ત્રિકોણમિતીઉઅ ગુણોત્તર મેળવો. 

ખૂણા Bના ત્રિકોણમિતીય ગુણોત્તર મેળવવા માટે આપણે સૌપ્રથમ ત્રિકોણની ત્રીજી બાજુ ABનું માપ મેળવવુંં પડશે. પાયથાગોરસના પ્રમેયથી આપણે ત્રિજીબાજુનું માપ મેળવીએ. 



AB2 + AC2 = BC2 

∴ AB2 = BC2 - AC

=(13)2 - (5)2 

=169 - 25 = 144 

∴ AB = square root of bold 144 =12

હવે, ત્રિકોણમિતીય ગુણોત્તરની વ્યાખ્યા પરથી, 

bold sin bold space bold B bold space bold equals bold space bold AC over bold BC bold space bold equals bold space bold 5 over bold 13 bold comma bold space bold cos bold space bold B bold space bold equals bold AB over bold BC bold space bold equals bold space bold 12 over bold 13

bold ten bold space bold B bold space bold equals bold space bold AC over bold AB bold space bold equals bold space bold 5 over bold 12 bold comma bold space bold cot bold space bold B bold space bold equals bold space bold AB over bold AC bold space bold equals bold space bold 12 over bold 13

bold sec bold space bold B bold space bold equals bold space bold BC over bold AB bold space bold equals bold space bold 13 over bold 12 bold comma bold space bold cosec bold space bold B bold space bold equals bold space bold BC over bold AC bold space bold equals bold space bold 13 over bold 5

નોંધ : કાટકોણ ત્રિકોણમાં કર્ણ સૌથી મોટી બાજુ હોવાથી sin A અને cos Aનું માપ હંમેશા 1 થી ઓછું થશે. વળી, ધન સંખ્યાઓનો ગુણોત્તર હોવાથી તે ધન થશે. આથી 0 < sin A < 1, 0 < cos A < 1. 

ABCમાં, B કાટકોણ હોય, BC = 7 અને AC - AB = 1, તો sin C તથા cos C શોધો. 


ABCમાં, AC - AB = 1

∴ AC = AB + 1 

કાટકોણ ∆ABCમાં m∠B = 90, 



AC2 = AB2 + BC2 

∴ (AB + 1)2 = AB2 + BC2 

∴ 1 + 2AB + AB2 = AB2 + BC2 

∴1 + 2AB = BC

∴ 2AB = 72 - 1       (BC = 7) 

∴ 2AB = 48 

∴ AB = 24 અને AC = 1 +AB = 25 

તેથી sin C = bold AB over bold AC bold space bold equals bold space bold 24 over bold 25 અને cos C= bold BC over bold AC bold space bold equals bold space bold 7 over bold 25