Question #28e21

3 Answers
Aug 10, 2017

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

Aug 10, 2017

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")

Aug 10, 2017

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.