Identities of cos(x) and sin(x) | ||||||||||||
Find x if : 1/sin(x)-1/cos(x)=2*sqrt(2) Solution : Squared for both sides : [1/sin(x)-1/cos(x)]^2=8 [cos(x)-sin(x)]^2/[sin(x)*cos(x)]^2=8 [cos^2(x)-2*cos(x)*sin(x)+sin^2(x)]/[2*sin(x)*cos(x)/2]^2=8 [1-sin(2x)]/[sin^2(2x)/4]=8 4*[1-sin(2x)]/sin^2(2x)=8 divide both sides by 4 : [1-sin(2x)]/sin^2(2x)=2 multiply both sides by sin^2(2x) : 1-sin(2x)=2*sin^2(2x) 2*sin^2(2x)+sin(2x)-1=0 2*sin^2(2x)+2*sin(2x)-sin(2x)-1=0 2*sin(2x)*[sin(2x)+1]-[sin(2x)+1]=0 [sin(2x)+1]*[2*sin(2x)-1]=0 sin(2x)+1=0 sin(2x)=-1 2x=270+k*360 if k=0 then 2x=270 x1=135 deg if k=1 then 2x=630 x2=315deg 2sin(2x)-1=0 sin(2x)=1/2 2x=30+k*360 k=0 then 2x=30 x3=15deg k=1 then 2x=390 x4=195 deg 2x=150+k*360 k=0 then 2x=150 x5=75 k=1 then 2x=510 x6=255deg |
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