Find the tangent of an angle on the unit circle: $ heta = frac{5pi}{4} $

To find $ \tan(\theta) $ where $ \theta = \frac{5\pi}{4} $:

First, recognize that $ \frac{5\pi}{4} $ is in the third quadrant of the unit circle.

In the third quadrant, the tangent function is positive.

The reference angle for $ \frac{5\pi}{4} $ is:

$ \pi – \frac{5\pi}{4} = \frac{\pi}{4} $

Using the reference angle, we have:

$ \tan(\frac{\pi}{4}) = 1 $

Thus, $ \tan(\frac{5\pi}{4}) = 1 $

We need to determine $ an( heta) $, where $ heta = frac{5pi}{4} $:

Locate $ frac{5pi}{4} $ on the unit circle, it is in the third quadrant.

The tangent function in the third quadrant is positive.

Calculate the reference angle:

$ frac{5pi}{4} – pi = frac{pi}{4} $

Since:

$ an(frac{pi}{4}) = 1 $

It follows that:

$ an(frac{5pi}{4}) = 1 $

The tangent of $ heta = frac{5pi}{4} $ is found as follows:

Since $ frac{5pi}{4} $ is in the third quadrant, where tangent is positive, and the reference angle is $ frac{pi}{4} $:

$ an(frac{5pi}{4}) = an(frac{pi}{4}) = 1 $

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