Solve the equation on the interval $$ 0 \leq heta

Question

Solve the equation on the interval $$ 0 \leq 	heta<2 \pi $$ $$ \cos \left(2 	heta-\frac{\pi}{2}ight)=-1 $$ What are the solutions in the interval $$ 0 \leq 	heta<2 \pi $$ ? Select the correct choice and fill in any answer boxes in your choice below. A. The solution set is \{ \}. (Simplify your answer. Type an exact answer, using $$ \pi $$ as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. There is no solution.

UpStudy AI · Accepted Answer

To solve the equation $$ \cos \left(2 \theta - \frac{\pi}{2}\right) = -1 $$ for $$ \theta $$ in the interval $$ 0 \leq \theta < 2\pi $$, follow these steps:

1. **Identify When Cosine Equals -1:**
   The cosine function equals $$-1$$ at angles of the form:
   $$
   2\theta - \frac{\pi}{2} = \pi + 2\pi k \quad \text{for integer } k
   $$
   
2. **Solve for $$ \theta $$:**
   $$
   2\theta = \pi + \frac{\pi}{2} + 2\pi k = \frac{3\pi}{2} + 2\pi k
   $$
   $$
   \theta = \frac{3\pi}{4} + \pi k
   $$
   
3. **Find Solutions Within the Given Interval:**
   - For $$ k = 0 $$:
     $$
     \theta = \frac{3\pi}{4}
     $$
   - For $$ k = 1 $$:
     $$
     \theta = \frac{3\pi}{4} + \pi = \frac{7\pi}{4}
     $$
   - For $$ k = 2 $$ or any other integer, $$ \theta $$ falls outside the interval $$ 0 \leq \theta < 2\pi $$.
   
4. **Conclusion:**
   The solutions within the interval $$ 0 \leq \theta < 2\pi $$ are:
   $$
   \left\{ \frac{3\pi}{4}, \frac{7\pi}{4} \right\}
   $$

**Answer:**

**A.** The solution set is $$\left\{\, \dfrac{3\pi}{4},\; \dfrac{7\pi}{4} \,\right\}$$.
The solutions are $$ \theta = \frac{3\pi}{4} $$ and $$ \theta = \frac{7\pi}{4} $$.

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