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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].
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].
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