Question

Question #4d2b7

Answer 1

C)

Explanation:

"Brute Force" method just sets the expressions equal and checks the values. There may be a more refined trig way to manipulate the expressions, but this is simplest for multiple-choice.
`12sin^2x = 1 - sinx`

`sin^2x = (1 - sinx)/12` (easier to calc/compare from the sine vales)

`x = 194.5` or `344.5`
`sinx = -0.25` ; `sin^2x = 0.06`

`0.063 = 0.75/12 = 0.63`

Answer 2

`(B)`

Explanation:

`12sin2x+sinx-1=0`

`sin2x=2sinxcosx`

`24sinxcosx+sinx-1=0`

`cosx=(1-sinx)/(24sinx)`

`cos^2x=(1-sinx)^2/(576sin^2x)`

`1-sin^2x=(1+sin^2x-2sinx)/(576sin^2x)`

`576sin^2x(1-sin^2x)=1+sin^2x-2sinx`

`576sin^2x-576sin^4x-1-sin^2x+2sinx=0`

`-576sin^4x+575sin^2x+2sinx-1=0`

Now you would have to solve for `sinx` which is not easy from this equation of 4th degree.

I think you meant:

`12sin^2x+sinx-1=0`

You can use the quadratic formula to solve for `sinx`:

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`sinx=(-1+-sqrt(1^2-4(12)(-1)))/(2(12)`

`sinx=(-1+-7)/24`

One answer is:

`sinx=-1/3`

`x=arcsin(-1/3)=-19.47 Degrees` which is answer `(B)`

The second answer is:

`sinx=1/4`

`x=arcsin(1/4)=14.5 Degrees`

Even though this matches answer `(A)` it is not correct because it does not fall in the given interval.

Answer `(B)` is the only correct answer.