How to evaluate arcsin?
Understand the Problem
The question is asking for a method or process to evaluate the arcsine function, which is the inverse of the sine function. This typically involves finding an angle whose sine is a given value and can also involve understanding the range and domain of the arcsine function.
Answer
$$ y = \frac{\pi}{6} $$
Answer for screen readers
$$ y = \frac{\pi}{6} $$
Steps to Solve
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Identify the value for arcsine
Determine the specific value of sine you are interested in. For example, if you want to find $y$ such that $\sin(y) = \frac{1}{2}$, the value to evaluate is $\frac{1}{2}$. -
Remember the range of arcsine
The arcsine function, denoted as $\arcsin(x)$, has specific limits: its range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$ or in degrees, $[-90^{\circ}, 90^{\circ}]$. It means the output angle will always be within this interval. -
Find the corresponding angle
You need to find the angle $y$ such that $\sin(y)$ equals your initial value. For instance, if your value is $\frac{1}{2}$, then the angle $y$ that satisfies this within the specified range is $\frac{\pi}{6}$ (or $30^{\circ}$). -
Write the relation
Using the arcsine function, your result can be expressed as: $$ y = \arcsin\left(\frac{1}{2}\right) $$ -
Evaluate arcsine
Thus, evaluate the arcsine to find the angle: $$ y = \frac{\pi}{6} $$
$$ y = \frac{\pi}{6} $$
More Information
The arcsine function is crucial in trigonometry, allowing us to find angles when given the sine values. The value $\frac{\pi}{6}$ corresponds to an angle of $30^{\circ}$, indicating that the sine of $30^{\circ}$ is $\frac{1}{2}$.
Tips
- Forgetting the range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$ can lead to choosing incorrect angles. Always ensure the angle is between these limits.
- Misinterpreting the sine function can result in incorrect values. Always verify that the sine of your resulting angle matches the initial value.
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