Question

`1/costheta+3sinthetatantheta+4=0` can be expressed as `3cos^2theta - 4costheta -4=0`. Hence solve the equation `1/costheta+3sinthetatantheta+4=0` for `0<=theta<=360` ?

Answer 1

`x=131.8^@" or "x=228.2^@`

Explanation:

`1/costheta+3sinthetaxxsintheta/costheta+4=0`

`rArr1/costheta+(3sin^2theta)/costheta+(4costheta)/costheta=0`

`rArr(1+3(1-cos^2theta)+4costheta)/costheta=0`

`rArr(-3cos^2theta+4costheta+4)/costheta=0"`

`rArr-3cos^2theta+4costheta+4=0larrcolor(blue)"multiply by - 1"`

`rArr3cos^2theta-4costheta-4=0`

`"using "3cos^2theta-4costheta-4=0" to solve"`

`3cos^2theta+2costheta-6costheta-4=0larr"split "costheta" term"`

`color(red)(costheta)(3costheta+2)color(red)(-2)(3costheta+2)=0larr"grouping"`

`rArr(costheta+2)(color(red)(3costheta+2))=0`

`"equate each factor to zero and solve for "theta`

`costheta+2=0tocostheta=-2larrcolor(red)"no solution"`

`3costheta+2=0tocostheta=-2/3`

`costheta<0rArrthetacolor(blue)" in second/third quadrants"`

`theta=cos^-1(2/3)=48.2^@larrcolor(red)"related acute angle"`

`rArrtheta=(180-48.2)^@=131.8^@`

`rArrtheta=(180+48.2)^@=228.2^@`

Answer 2

`1/costheta+3sinthetatantheta+4=0`

`=>1/costheta+3sintheta*sintheta/costheta+4=0`

`=>1+3sin^2theta+4costheta=0`

`=>1+3(1-cos^2theta)+4costheta=0`

`=>3cos^2theta-4costheta-4=0`

`=>3cos^2theta-6costheta+2costheta-4=0`

`=>3costheta(costheta-2)+2(costheta-2)=0`

`=>(3costheta+2)(costheta-2)=0`

`costheta=2` not possible

So `costheta=-2/3=cos131.8^@=cos(360-131.8)^@`

Hence

`theta=131.8^@ and theta=228.2^@`