Find the value of $ arcsin(frac{1}{2}) $ in radians using the unit circle.

To find the value of $ \arcsin(\frac{1}{2}) $, consider the unit circle and the definition of arcsin. The arcsin function outputs the angle whose sine is the given value within the range $ -\frac{\pi}{2} $ to $ \frac{\pi}{2} $.

For $ \arcsin(\frac{1}{2}) $, we need to find the angle $ \theta $ such that $ \sin(\theta) = \frac{1}{2} $. On the unit circle, $ \sin(30^{\circ}) = \frac{1}{2} $ or equivalently, in radians:

$ \theta = \frac{\pi}{6} $

Thus, the value of $ \arcsin(\frac{1}{2}) $ is:

$ \arcsin(\frac{1}{2}) = \frac{\pi}{6} $

To find the angle represented by $ arcsin(-frac{1}{2}) $, look at the unit circle. The arcsin function returns an angle between $ -frac{pi}{2} $ and $ frac{pi}{2} $ where the sine of the angle equals the input value.

For $ arcsin(-frac{1}{2}) $, we find the angle $ heta $ where $ sin( heta) = -frac{1}{2} $. On the unit circle, $ sin(-30^{circ}) = -frac{1}{2} $, or in radians:

$ heta = -frac{pi}{6} $

Thus, the value of $ arcsin(-frac{1}{2}) $ is:

$ arcsin(-frac{1}{2}) = -frac{pi}{6} $

To find $ arcsin(1) $, we need the angle $ heta $ such that $ sin( heta) = 1 $ within $ -frac{pi}{2} $ to $ frac{pi}{2} $.

On the unit circle, $ sin(frac{pi}{2}) = 1 $:

$ arcsin(1) = frac{pi}{2} $

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