Find the value of $ an(135^circ) $ using the unit circle

To find the value of $ \tan(135^\circ) $ using the unit circle, we need to recall that $ \tan\theta $ is the ratio of the y-coordinate to the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

The angle $ 135^\circ $ is in the second quadrant, where the tangent is negative. It corresponds to the reference angle $ 45^\circ $.

For $ 45^\circ $, the coordinates on the unit circle are:

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

In the second quadrant, the x-coordinate is negative, so the point is:

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

Thus,

$ \tan(135^\circ) = \frac{\frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = -1 $

To find $ an(135^circ) $, recall that the unit circle helps determine trigonometric values based on angles.

The angle $ 135^circ $ is in the second quadrant, and its reference angle is $ 45^circ $.

Coordinates of $ 45^circ $ are:

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

In the second quadrant, x is negative:

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

Therefore,

$ an(135^circ) = frac{frac{sqrt{2}}{2}}{- frac{sqrt{2}}{2}} = -1 $

To find $ an(135^circ) $, use the unit circle:

The coordinates are:

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

So,

$ an(135^circ) = -1 $

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