Question
There is no solution to the equation \( \csc x=-1 \). A. True B. False
Ask by Hamilton Schultz. in the United States
Jan 30,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equation \( \csc x = -1 \) has solutions, so the statement is false.
Solution
The statement claims that there is no solution to the equation \( \csc x = -1 \).
Let's analyze the equation:
\[
\csc x = -1 \quad \Rightarrow \quad \frac{1}{\sin x} = -1 \quad \Rightarrow \quad \sin x = -1
\]
The equation \( \sin x = -1 \) has solutions. Specifically, \( \sin x = -1 \) at:
\[
x = \frac{3\pi}{2} + 2k\pi \quad \text{for any integer } k
\]
Therefore, the original statement is **False**.
**Answer:**
B. False
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To determine if the statement is true or false, let's analyze the equation \( \csc x = -1 \). The cosecant function is defined as the reciprocal of the sine function, so \( \csc x = -1 \) translates to \( \sin x = -1 \). The sine function equals -1 at specific angles, most notably at \( x = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer. Therefore, there are indeed solutions, making the statement false. B. False
