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Solve −4sin^2(x) − 3cos(x) = −3

The smallest non-negative radian solution is: _____


The next smallest non-negative radian solution is: _____


Thank you!

1 Answer

3 votes

Answer:

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.

User Nithin Bhaskar
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