Calculate the value of $ an(frac{7pi}{4})$ and find the reference angle.

First, let’s determine the reference angle for $\frac{7\pi}{4}$. We know that:

$\frac{7\pi}{4} = 2\pi – \frac{\pi}{4}$

So, the reference angle is:

$\frac{\pi}{4}$

Next, we find the value of $\tan(\frac{7\pi}{4})$. Since $\frac{7\pi}{4}$ is in the fourth quadrant, and the tangent function is positive in the fourth quadrant, we have:

$\tan(\frac{7\pi}{4}) = -\tan(\frac{\pi}{4})$

We know that:

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

Therefore:

$\tan(\frac{7\pi}{4}) = -1$

First, convert $frac{7pi}{4}$ to degrees:

$frac{7pi}{4} imes frac{180}{pi} = 315^circ$

The reference angle is:

$360^circ – 315^circ = 45^circ$

Thus, the reference angle is $45^circ$ or $frac{pi}{4}$. Now, since $315^circ$ is in the fourth quadrant, where tangent is negative:

$ an(315^circ) = – an(45^circ)$

We know that:

$ an(45^circ) = 1$

So:

$ an(315^circ) = -1$

We know that:

$frac{7pi}{4} = 2pi – frac{pi}{4}$

So the reference angle is $frac{pi}{4}$. In the fourth quadrant, tangent is negative:

$ an(frac{7pi}{4}) = -1$

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