To solve the equation [tex]\( -5 \sin^2(x) - 4 \cos(x) + 4 = 0 \)[/tex], follow these steps:
1. Express [tex]\(-5 \sin^2(x)\)[/tex] in terms of [tex]\(\cos(x)\)[/tex]:
Recall the Pythagorean identity:
[tex]\[
\sin^2(x) = 1 - \cos^2(x)
\][/tex]
Substitute this into the equation:
[tex]\[
-5 (1 - \cos^2(x)) - 4 \cos(x) + 4 = 0
\][/tex]
Simplify:
[tex]\[
-5 + 5 \cos^2(x) - 4 \cos(x) + 4 = 0
\][/tex]
[tex]\[
5 \cos^2(x) - 4 \cos(x) - 1 = 0
\][/tex]
2. Solve the quadratic equation for [tex]\(\cos(x)\)[/tex]:
Let [tex]\( u = \cos(x) \)[/tex]. The equation becomes:
[tex]\[
5u^2 - 4u - 1 = 0
\][/tex]
Solve this quadratic equation using the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[
u = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 5 \cdot (-1)}}{2 \cdot 5}
\][/tex]
[tex]\[
u = \frac{4 \pm \sqrt{16 + 20}}{10}
\][/tex]
[tex]\[
u = \frac{4 \pm \sqrt{36}}{10}
\][/tex]
[tex]\[
u = \frac{4 \pm 6}{10}
\][/tex]
Therefore, we have two solutions for [tex]\( u \)[/tex]:
[tex]\[
u = \frac{10}{10} = 1
\][/tex]
and
[tex]\[
u = \frac{-2}{10} = -0.2
\][/tex]
So, [tex]\( \cos(x) = 1 \)[/tex] or [tex]\( \cos(x) = -0.2 \)[/tex].
3. Find the corresponding values of [tex]\( x \)[/tex] for each solution:
- For [tex]\( \cos(x) = 1 \)[/tex]:
[tex]\[
x = 0 \quad (\text{since } \cos(0) = 1)
\][/tex]
- For [tex]\( \cos(x) = -0.2 \)[/tex]:
Solve for [tex]\( x \)[/tex] within the range [tex]\( [0, 2\pi] \)[/tex]. The solutions are:
[tex]\[
x = \arccos(-0.2) \quad \text{and} \quad x = 2\pi - \arccos(-0.2)
\][/tex]
Calculating [tex]\( \arccos(-0.2) \)[/tex]:
[tex]\[
x \approx 1.772154 \quad (\text{radians})
\][/tex]
[tex]\[
2\pi - \arccos(-0.2) \approx 4.511032 \quad (\text{radians})
\][/tex]
4. Identify the smallest non-negative radian solutions:
The smallest solution:
[tex]\[
x = 0
\][/tex]
The next smallest solution:
[tex]\[
x \approx 1.772154
\][/tex]
Therefore, the smallest non-negative radian solution is [tex]\( \boxed{0.0} \)[/tex], and the next smallest non-negative radian solution is [tex]\( \boxed{1.7721542475852274} \)[/tex].
