Find the values of $ heta $ for which $ an( heta) = 1 $ in the unit circle

To find the values of $ \theta $ for which $ \tan(\theta) = 1 $ on the unit circle, we need to identify the angles where the tangent function is equal to 1.

The tangent function is defined as the ratio of the sine and cosine functions:

$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $

For $ \tan(\theta) = 1 $, we have:

$ \frac{\sin(\theta)}{\cos(\theta)} = 1 $

This implies:

$ \sin(\theta) = \cos(\theta) $

On the unit circle, this equality occurs at:

$ \theta = \frac{\pi}{4} + k\pi $

where $ k $ is any integer. Therefore, the solutions are:

$ \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \ldots $

To find the values of $ heta $ for which $ an( heta) = 1 $ on the unit circle, we note that the tangent function is the ratio of the sine and cosine functions:

$ an( heta) = frac{sin( heta)}{cos( heta)} $

When $ an( heta) = 1 $, it implies:

$ sin( heta) = cos( heta) $

This occurs at:

$ heta = frac{pi}{4} + kpi $

where $ k $ is any integer. Therefore, the solutions are:

$ heta = frac{pi}{4}, frac{5pi}{4}, frac{9pi}{4}, ldots $

The values of $ heta $ for which $ an( heta) = 1 $ on the unit circle are:

$ heta = frac{pi}{4} + kpi $

where $ k $ is any integer.

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