Questions: Determine a general formula (or formulas) for the solution to the following equation. Then, determine the specific solutions (if any) on the interval [0,2 π ). cos θ+2 cos^2 θ=0 Describe in the most concise way possible the general solution to the given equation, where k is any integer. Select the correct choice below and fill in all the answer boxes within your choice. (Simplify your answers. Type any angle measures in radians. Use the smallest non-negative angle possible when describing the general form of each angle. Use ascending order, entering the expression based on the smallest angle first. Type exact answers, using π as needed. Use integers or fractions for any numbers in the expressions.) A. θ= + k B. θ= + k or θ= + k C. θ= + k or θ= + k or θ= + k D. θ= + k or θ= + k or θ= + k or θ= + k E. There is no solution.

Determine a general formula (or formulas) for the solution to the following equation. Then, determine the specific solutions (if any) on the interval [0,2 π ). cos θ+2 cos^2 θ=0 Describe in the most concise way possible the general solution to the given equation, where k is any integer. Select the correct choice below and fill in all the answer boxes within your choice. (Simplify your answers. Type any angle measures in radians. Use the smallest non-negative angle possible when describing the general form of each angle. Use ascending order, entering the expression based on the smallest angle first. Type exact answers, using π as needed. Use integers or fractions for any numbers in the expressions.) A. θ= + k B. θ= + k or θ= + k C. θ= + k or θ= + k or θ= + k D. θ= + k or θ= + k or θ= + k or θ= + k E. There is no solution.
Transcript text: Determine a general formula (or formulas) for the solution to the following equation. Then, determine the specific solutions (if any) on the interval $[0,2 \pi$ ). \[ \cos \theta+2 \cos ^{2} \theta=0 \] Describe in the most concise way possible the general solution to the given equation, where k is any integer. Select the correct choice below and fill in all the answer boxes within your choice. (Simplify your answers. Type any angle measures in radians. Use the smallest non-negative angle possible when describing the general form of each angle. Use ascending order, entering the expression based on the smallest angle first. Type exact answers, using $\pi$ as needed. Use integers or fractions for any numbers in the expressions.) A. $\theta=$ $\square$ $+$ k B. $\theta=$ $\square$ $+\square$ k or $\theta=$ $\square$ $+$ k C. $\theta=$ $\square$ $+$ $\square$ k or $\theta=$ $\square$ $+$ $\square$ $k$ or $\theta=$ $\square$ $+$ k D. $\theta=$ $\square$ $+$ $k$ or $\theta=$ $\square$ $+\square$ $\square$ $k$ or $\theta=$ $\square$ $+$ $k$ or $\theta=$ $\square$ $+\square$ k E. There is no solution.
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Solution

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Solution Steps

To solve the equation \(\cos \theta + 2 \cos^2 \theta = 0\), we can start by factoring the equation. This involves recognizing that it can be rewritten in a form that allows us to use the zero-product property. Once factored, we can solve for \(\theta\) by finding the angles that satisfy each factor. Finally, we determine the specific solutions within the interval \([0, 2\pi]\).

Step 1: Rewrite the Equation

We start with the equation: \[ \cos \theta + 2 \cos^2 \theta = 0 \] This can be factored as: \[ \cos \theta (1 + 2 \cos \theta) = 0 \]

Step 2: Solve Each Factor

From the factored equation, we have two cases to solve:

  1. \(\cos \theta = 0\)
  2. \(1 + 2 \cos \theta = 0\)
Case 1: \(\cos \theta = 0\)

The solutions for \(\cos \theta = 0\) are: \[ \theta = \frac{\pi}{2} + k\pi \quad \text{for any integer } k \] Within the interval \([0, 2\pi]\), the specific solutions are: \[ \theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2} \]

Case 2: \(1 + 2 \cos \theta = 0\)

Rearranging gives: \[ \cos \theta = -\frac{1}{2} \] The solutions for \(\cos \theta = -\frac{1}{2}\) are: \[ \theta = \frac{2\pi}{3} + 2k\pi \quad \text{and} \quad \theta = \frac{4\pi}{3} + 2k\pi \quad \text{for any integer } k \] Within the interval \([0, 2\pi]\), the specific solutions are: \[ \theta = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3} \]

Step 3: Compile All Solutions

Combining all specific solutions from both cases, we have: \[ \theta = \frac{\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{3\pi}{2} \]

Final Answer

The specific solutions to the equation \(\cos \theta + 2 \cos^2 \theta = 0\) in the interval \([0, 2\pi]\) are: \[ \boxed{\left\{ \frac{\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{3\pi}{2} \right\}} \]

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