Solve the equation on the interval [tex]0 \leq \theta \ \textless \ 2\pi[/tex].

[tex]\sin^2 \theta = 4(\cos(-\theta) - 1)[/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The solution set is [tex]\{\square \}[/tex].
(Simplify your answer. Type an exact answer, using [tex]\pi[/tex] as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

B. There is no solution.



Answer :

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].