What is the sine inverse of 1?
Understand the Problem
The question is asking for the inverse sine (also known as arcsine) of the value 1, which involves finding the angle whose sine is 1.
Answer
$$ \frac{\pi}{2} $$
Answer for screen readers
The final answer is
$$ \frac{\pi}{2} $$
Steps to Solve
- Identify the function We are looking for the inverse sine, or arcsine, of 1, which can be written as:
$$ \arcsin(1) $$
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Understand the range of arcsine The arcsine function has a range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$. This means the output of $\arcsin(x)$ will always be between these two angles.
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Find the angle Now, we need to determine which angle within the range produces a sine of 1. We know from the unit circle that:
$$ \sin\left(\frac{\pi}{2}\right) = 1 $$
Thus, the angle we are looking for is
$$ \frac{\pi}{2} $$
Therefore, we can conclude:
$$ \arcsin(1) = \frac{\pi}{2} $$
The final answer is
$$ \frac{\pi}{2} $$
More Information
The sine function reaches 1 at an angle of $\frac{\pi}{2}$ radians (or 90 degrees), which is a significant angle in trigonometry. The arcsine function is essential in many fields, including physics and engineering, where angle calculations are necessary.
Tips
- Confusing the range: A common mistake is to forget that the output of the arcsine function is restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Always remember to check that the angle you find fits within this range.
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