Mathematics
savianp
3

Find all solutions of the equation in the interval [0, 2pi]

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(1) Answers
katiebabyyy

[latex]sec^2(y)+6*tan(y)+4=0[/latex] first we have to change this [latex]sec^2(y)[/latex] we know this equation [latex]sin^2(y)+cos^2(y)=1[/latex] then, let's divide all members by [latex]cos^2(y)[/latex] [latex]\frac{sin^2(y)}{cos^2(y)}+\frac{cos^2(y)}{cos^2(y)}=\frac{1}{cos^2(y)}[/latex] then got... [latex]tan^2(y)+1=sec^2(y)[/latex] now we can replace it at original equation [latex]tan^2(y)+1+6*tan(y)+4=0[/latex] [latex]tan^2(y)+6*tan(y)+5=0[/latex] now we have to make another substitution... [latex]tan(y)=x[/latex] so [latex]x^2+6*x+5=0[/latex] then we found [latex]x_1=-5~~and~~x_2=-1[/latex] now we have to back to tan(y) [latex]tan(y)=-1[/latex] [latex]y=arctan(-1)[/latex] [latex]y=\frac{3\pi}{4}~~and~~y=\frac{7\pi}{4}[/latex] if you have a scientific calculator you'll find out this value to [latex]tan(y)=-5[/latex] in degrees [latex]y\approx281.31\º~~and~~y\approx101.31\º[/latex]

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