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

Answer:

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

Step-by-step 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

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

Where

$y_{1}=\frac{-3+5}{8}=\frac{2}{8}=\frac{1}{4}\\ y_{2}=\frac{-3-5}{8}=\frac{-8}{8}=-1$

But, $y=cos(x)$

So,

$cos(x)=\frac{1}{4}\\x=cos^{-1}(\frac{1}{4} ) \approx 1.32$ 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.