Find the coordinates on the unit circle where $ an( heta) = 1 $

To find the coordinates on the unit circle where $ \tan(\theta) = 1 $, we need to determine the angles $\theta $ for which this condition holds. We know that:

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

For the tangent to be 1, the sine and cosine must be equal. This occurs at angles:

$ \theta = \frac {\pi}{4} \text{ and } \theta = \frac {5\pi}{4} $

Now, we find the coordinates on the unit circle for these angles:

$ \text{At } \theta = \frac {\pi}{4}, \text{ the coordinates are } \left( \frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2} \right) $

$ \text{At } \theta = \frac {5\pi}{4}, \text{ the coordinates are } \left( -\frac {\sqrt {2}}{2}, -\frac {\sqrt {2}}{2} \right) $

Thus, the coordinates on the unit circle where $ \tan(\theta) = 1 $ are:

$ \left( \frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2} \right) $

and

$ \left( -\frac {\sqrt {2}}{2}, -\frac {\sqrt {2}}{2} \right) $

To find the coordinates on the unit circle where $ an( heta) = 1 $, we use the fact that:

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

This occurs at:

$ heta = frac {pi}{4} ext{ and } heta = frac {5pi}{4} $

Therefore, the coordinates are:

$ left( frac {sqrt {2}}{2}, frac {sqrt {2}}{2}
ight) $

and

$ left( -frac {sqrt {2}}{2}, -frac {sqrt {2}}{2}
ight) $

The coordinates on the unit circle where $ an( heta) = 1 $ are found at:

$ heta = frac {pi}{4} $

and

$ heta = frac {5pi}{4} $

Thus, the coordinates are:

$ left( frac {sqrt {2}}{2}, frac {sqrt {2}}{2}
ight) $

and

$ left( -frac {sqrt {2}}{2}, -frac {sqrt {2}}{2}
ight) $

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