To solve the equation [tex]\(\sin^2 \theta = 4(\cos(-\theta)-1)\)[/tex] in the interval [tex]\(0 \leq \theta < 2\pi\)[/tex], we follow these steps:
1. Simplify the equation:
[tex]\[
\sin^2 \theta = 4(\cos(-\theta) - 1)
\][/tex]
2. Use the property of the cosine function:
Since [tex]\(\cos(-\theta) = \cos(\theta)\)[/tex], the equation simplifies to:
[tex]\[
\sin^2 \theta = 4(\cos \theta - 1)
\][/tex]
3. Express in terms of sine and cosine:
Recall that [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]. Substituting [tex]\(\sin^2 \theta = 1 - \cos^2 \theta\)[/tex], the original equation becomes:
[tex]\[
1 - \cos^2 \theta = 4(\cos \theta - 1)
\][/tex]
4. Reorganize the equation:
Combine like terms to obtain a quadratic equation in terms of [tex]\(\cos \theta\)[/tex]:
[tex]\[
1 - \cos^2 \theta = 4\cos \theta - 4
\][/tex]
Rearrange all terms to one side:
[tex]\[
\cos^2 \theta + 4\cos \theta - 5 = 0
\][/tex]
5. Solve the quadratic equation:
Let [tex]\(x = \cos \theta\)[/tex]. The quadratic equation becomes:
[tex]\[
x^2 + 4x - 5 = 0
\][/tex]
6. Factor the quadratic equation:
[tex]\[
(x + 5)(x - 1) = 0
\][/tex]
Hence, the solutions are:
[tex]\[
x = -5 \quad \text{or} \quad x = 1
\][/tex]
Since [tex]\(\cos \theta\)[/tex] must be in the range [tex]\([-1, 1]\)[/tex], [tex]\(x = -5\)[/tex] is not valid.
7. Find [tex]\(\theta\)[/tex] for valid [tex]\(\cos \theta\)[/tex]:
The valid solution is:
[tex]\[
\cos \theta = 1
\][/tex]
8. Determine [tex]\(\theta\)[/tex] in the given interval:
[tex]\(\cos \theta = 1\)[/tex] occurs when:
[tex]\[
\theta = 0
\][/tex]
9. Verify the solution in the given interval [tex]\(0 \leq \theta < 2\pi\)[/tex]:
The angle [tex]\(\theta = 0\)[/tex] falls within the interval [tex]\(0 \leq \theta < 2\pi\)[/tex].
Therefore, the solution set is:
[tex]\[
\{ \theta = 0 \}
\][/tex]
Hence, the correct choice is:
A. The solution set is [tex]\(\{0\}\)[/tex].
