Question

Question #28e21

Answer 1

`sinx=(4+cosx)/8 or sinx= 1/2+cosx/8`

Explanation:

`8sinx=4+cosx`

`=>`Dividing by `8` in both sides.

`=>(8sinx)/8=(4+cosx)/8`

`=>(cancelcolor(red)(8)sinx)/cancelcolor(red)(8)=(4+cosx)/8`

`=>sinx=(4+cosx)/8 or` On simplifying

`=>sinx=4/8+cosx/8`

`=>sinx=(1*4)/(2*4)+cosx/8`

`=>sinx=(1*cancelcolor(red)(4))/(2*cancelcolor(red)(4))+cosx/8`

`=>sinx=1/2+cosx/8`

So answer = `sinx=(4+cosx)/8 or sinx= 1/2+cosx/8`

Answer 2

`8sinx=4+cosx`

`=>(8sinx-4)^2=cos^2x`

`=>64sin^2x-64sinx+16=1-sin^2x`

`=>65sin^2x-64sinx+15=0`

`=>65sin^2x-39sinx-25sinx+15=0`

`=>13sinx(5sinx-3)-5(5sinx-3)=0`

`=>(5sinx-3)(13sinx-5)=0`

So `sinx =3/5 and sin x= 5/13`

When `sinx =3/5` then `cosx =pmsqrt(1-sin^2x)=pm4/5`

`sinx=3/5 and cosx=4/5` when #x
in " 1st quadrant "
If we put these two values in the given equation we get

`LHS=8sinx=8xx3/4=24/5`
and

`RHS=4+cosx=4+4/5=24/5`

Here `LHS=RHS`

So we can say that `sinx =3/5` satisfies the given equation and this value is an acceptable solution when angle x is in 1st quadrant,

If x is in 2nd quadrant the `sinx =3/5` but `cosx= -4/5`, these values do not satisfy the given equation

Again

When `sinx =5/13` then `cosx =pmsqrt(1-sin^2x)=pm12/13`

`sinx =5/13 and cosx =12/13` when `x in " 1st quadrant"`

If we put these two values in the given equation we get

`LHS=8sinx=8xx5/13=40/13`
and

`RHS=4+cosx=4+12/13=64/13`

Here `LHS!=RHS`

But if x is in 2nd quadrant the `sinx =5/13` but `cosx= -12/13`, these values do satisfy the given equation

Hence solution `color(red)(tosinx =3/5" when x is in 1st qudrant")`

and `color(red)(tosinx =5/13" when x is in 2nd quadrant")`

Answer 3

`sinx=3/5, or, 5/13.`

Explanation:

Given that, `8sinx=4+cosx.`

`:. 8sinx-4=cosx.`

`:. (8sinx-4)^2=cos^2x=1-sin^2x.`

`:. 64sin^2x-64sinx+16+sin^2x-1=0, i.e.,`

`65sin^2x-64sinx+15=0.`

Applying the Quadr. Formula, we get,

`sinx={64+-sqrt(64^2-4*65*15)}/(2*65),`

`=(64+-sqrt(4096-3900))/130=(64+-sqrt196)/130=(64+-14)/130.`

`rArr sinx=78/130, or, 50/130.`

`:. sinx=3/5, or, 5/13.`