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# How do you find the exact value of tan 405 degrees?

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Jun 11, 2018

#### Answer:

$\tan 405 = 1$

#### Explanation:

On the unit circle, the nearest $x$-axis to $405$ degrees is $360$ degrees. This means the reference angle (difference between the two) is $45$ degrees.

What do we know about $45$ degrees?

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

where the $x$ coordinate is the $\cos$ value and the $y$ coordinate is the $\sin$ value. We understand $\tan \theta$ to be defined as

$\sin \frac{\theta}{\cos} \theta$

Thus, we have $\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$, which simplifies to $1$.

We found $\tan 45$, which is the same as $\tan 405$ because $45$ degrees is its reference angle.

$\tan 405 = 1$

Hope this helps!

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