Answered

Which of the following is not a value of $\theta$ when $\sin \theta=-1$?


A$\pi$
B$-\frac{\pi}{2}$
C$\frac{7 \pi}{2}$
D$\frac{3 \pi}{2}$

Verified Answer :

  • The equation \(\sin \theta = -1\) is satisfied when \(\theta = \frac{3\pi}{2} + 2k\pi\) for any integer \(k\).
  • Testing each value:
  • \(\sin(\pi) = 0 eq -1\)
  • \(\sin(-\frac{\pi}{2}) = -1\)
  • \(\sin(\frac{7\pi}{2}) = \sin(\frac{3\pi}{2} + 2\pi) = -1\)
  • \(\sin(\frac{3\pi}{2}) = -1\)
  • Only \(\pi\) does not satisfy the equation.
  • Final Answer: \(\boxed{\pi}\)

Explanation

  1. Problem Analysis

The goal is to determine which of the given values for \(\theta\) does not satisfy the equation \(\sin \theta = -1\). We are given the set \(\{\pi, -\frac{\pi}{2}, \frac{7 \pi}{2}, \frac{3 \pi}{2}\}\). The sine function, \(\sin \theta\), evaluates to \(-1\) at specific points on the unit circle. Specifically, \(\sin \theta = -1\) if and only if \(\theta\) is of the form \(\frac{3\pi}{2} + 2k\pi\), where \(k\) is any integer. This corresponds to the point \((0, -1)\) on the unit circle.

  1. Testing the Options

We will test each option by checking if it satisfies the condition \(\sin \theta = -1\).

  1. For \(\theta = \pi\): \(\sin(\pi) = 0\). This is not equal to \(-1\).
  2. For \(\theta = -\frac{\pi}{2}\): \(\sin(-\frac{\pi}{2}) = -\sin(\frac{\pi}{2}) = -1\). This satisfies the equation.
  3. For \(\theta = \frac{7 \pi}{2}\): \(\frac{7 \pi}{2} = \frac{3 \pi}{2} + 2\pi\). Since \(\sin(\frac{3 \pi}{2} + 2\pi) = \sin(\frac{3 \pi}{2}) = -1\), this satisfies the equation.
  4. For \(\theta = \frac{3 \pi}{2}\): \(\sin(\frac{3 \pi}{2}) = -1\). This satisfies the equation.
  1. Conclusion

Based on our analysis, \(\sin(\pi) = 0\), which means \(\pi\) is not a solution to the equation \(\sin \theta = -1\). The other three values provided all result in a sine value of \(-1\). Thus, \(\pi\) is the correct choice.

Examples

Understanding the periodic nature of trigonometric functions is essential in physics and engineering. For example, when analyzing alternating current (AC) power or sound waves, we often need to determine specific time intervals where a voltage or air pressure reaches its minimum value. By identifying the angles where \(\sin \theta = -1\), one can predict exactly when these troughs occur, which is crucial for signal synchronization or structural vibration analysis.

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