Question

Solve −4sin^2(x) − 3cos(x) = −3

The smallest non-negative radian solution is: _____


The next smallest non-negative radian solution is: _____


Thank you!

Answer 1

The smallest radian solution is 3.14, approximately. And the next smallest radian solution is 1.32, approximately.

Explanation:

The given expression is

`-4sin^(2)(x)-3cos(x)=-3`

We know that
`sin^(2)(x)=1-cos^(2) (x)`

So,

`-4(1-cos^(2)(x))-3cos(x)=-3\\ -4+4cos^(2)(x)-3 cos(x)=-3\\4cos^(2)(x)-3cos(x)-4+3=0\\4cos^(2)(x)-3cos(x)-1=0`

Let's call
`y=cos(x)`, so

`4y^(2)-3y-1=0`

Where
`a=4`,
`b=-3` and
`c=-1`. Using the quadratic formula, we have

`y_(1,2)=\frac{-b(+-)\sqrt{b^(2)-4ac } }{2a}= \frac{-(-3)(+-)\sqrt{(-3)^(2)-4(4)(-1) } }{2(4)}\\y_(1,2)=(3(+-)√(9+16) )/(8)=(-3(+-)√(25) )/(8) =(-3(+-)5)/(8)`

Where

`y_(1)=(-3+5)/(8)=(2)/(8)=(1)/(4)\\ y_(2)=(-3-5)/(8)=(-8)/(8)=-1`

But,
`y=cos(x)`

So,

`cos(x)=(1)/(4)\\x=cos^(-1)((1)/(4) ) \approx 1.32` and
`cos(x)=-1}\\x=cos^(-1)(-1) \approx 3.14`

Therefore, the smallest radian solution is 3.14, approximately. And the next smallest radian solution is 1.32, approximately.