When Does Sine Theta Equal or Exceed One?

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In summary: Therefore, there are no values of $\theta$ that satisfy the given inequality, making it false for the empty set as well.In summary, the statement $|\sin{\theta}\ge 1|$ is always false, regardless of the set of values for $\theta$. Therefore, the correct answer is option e, the empty set.
  • #1
karush
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For $-\dfrac{\pi}{2}\le \theta \le \dfrac{\pi}{2}
\quad |\sin{\theta}\ge 1|$ is true for all and only the values of $\theta$ in which of the following sets
$a.\ \left\{-\dfrac{\pi}{2},\dfrac{\pi}{2}\right\}
\quad b.\ \left\{\dfrac{\pi}{2}\right\}
\quad c.\ \left\{\theta | -\dfrac{\pi}{2}< \theta < \dfrac{\pi}{2}\right\}
\quad d.\ \left\{\theta | -\dfrac{\pi}{2}\le \theta < \dfrac{\pi}{2} \le \right\}
\quad e.\ \textit{empty set}$

I chose b since $\sin\theta$ can only be 1 at $\dfrac{\pi}{2}$

I have seen a lot of students miss this one
not sure why

what is the symbol for an empty set? or better yet the latex code
 
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  • #2
I would like to clarify that the statement $|\sin{\theta}\ge 1|$ is not true for all values of $\theta$ in any of the given sets. The correct answer is actually option $e$, the empty set.

The reason for this is that the absolute value of $\sin{\theta}$ can never be greater than 1. In fact, it can only take values between 0 and 1 for any value of $\theta$. This means that the given inequality $|\sin{\theta}\ge 1|$ is always false, regardless of the set of values for $\theta$.

To further explain, let's consider each option:

a. The set $\left\{-\dfrac{\pi}{2},\dfrac{\pi}{2}\right\}$ only contains two values, which are the endpoints of the given interval. Plugging these values into the inequality, we get $|\sin{\left(-\dfrac{\pi}{2}\right)}|\ge 1$ and $|\sin{\left(\dfrac{\pi}{2}\right)}|\ge 1$. But the absolute value of $\sin{\left(-\dfrac{\pi}{2}\right)}$ and $\sin{\left(\dfrac{\pi}{2}\right)}$ are both equal to 1, not greater than 1. Therefore, the inequality is still false.

b. As mentioned in the forum post, the value $\dfrac{\pi}{2}$ is the only value where $\sin{\theta}$ can be equal to 1. However, the given inequality states that $|\sin{\theta}\ge 1|$, which means that the absolute value of $\sin{\theta}$ must be greater than 1. Since this is not possible, the inequality is still false.

c. This set contains all values of $\theta$ between $-\dfrac{\pi}{2}$ and $\dfrac{\pi}{2}$, excluding the endpoints. Again, plugging in any value from this set into the inequality will result in a false statement.

d. Similar to option c, this set also contains all values of $\theta$ between $-\dfrac{\pi}{2}$ and $\dfrac{\pi}{2}$, including the endpoints. But as we have seen before, the inequality is still false for these values.

e. The empty set, denoted by $\emptyset$,
 

Related to When Does Sine Theta Equal or Exceed One?

1. What does "sin theta >= 1" mean?

"Sin theta >= 1" means that the sine of an angle (represented by theta) is greater than or equal to 1. In other words, the value of the sine function at that angle is equal to or higher than 1.

2. Is it possible for the sine of an angle to be greater than or equal to 1?

No, it is not possible for the sine of an angle to be greater than 1. The maximum value of the sine function is 1, which occurs at 90 degrees or π/2 radians.

3. What is the significance of "sin theta >= 1" in trigonometry?

"Sin theta >= 1" is a mathematical inequality that can be used to solve trigonometric equations and inequalities. It also has practical applications in fields such as engineering, physics, and navigation.

4. How can I graph "sin theta >= 1"?

To graph "sin theta >= 1", you can plot points on a coordinate plane where the y-coordinate (sin theta) is equal to or greater than 1. This will result in a horizontal line at y = 1, passing through the points (0, 1) and (2π, 1).

5. Can you provide an example of a trigonometric equation involving "sin theta >= 1"?

One example of a trigonometric equation involving "sin theta >= 1" is sin(2θ) >= 1. This equation can be solved by finding the values of θ that make the sine of 2θ equal to or greater than 1.

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