Find the coordinates and trigonometric values of $frac{pi}{10}$ on the unit circle

To find the exact coordinates of $\frac{\pi}{10}$ on the unit circle, we need to compute both the cosine and sine of this angle.

First, recall that the unit circle is defined by the equation $x^2 + y^2 = 1$ where $x = \cos(\theta)$ and $y = \sin(\theta)$. For the angle $\theta = \frac{\pi}{10}$, we have:

$x = \cos\left(\frac{\pi}{10}\right)$

$y = \sin\left(\frac{\pi}{10}\right)$

Using the half-angle and product-to-sum identities, we find:

$\cos\left(\frac{\pi}{10}\right) = \sqrt{\frac{5 + \sqrt{5}}{8}}$

$\sin\left(\frac{\pi}{10}\right) = \sqrt{\frac{5 – \sqrt{5}}{8}}$

Therefore, the coordinates of $\frac{\pi}{10}$ on the unit circle are:

$\left(\sqrt{\frac{5 + \sqrt{5}}{8}}, \sqrt{\frac{5 – \sqrt{5}}{8}}\right)$

To determine the exact trigonometric values of $frac{pi}{10}$, we use trigonometric identities and properties of special angles.

From trigonometric tables or by using a calculator, it can sometimes be hard to find the precise values of $cosleft(frac{pi}{10}
ight)$ and $sinleft(frac{pi}{10}
ight)$. However, using advanced techniques, we find:

$cosleft(frac{pi}{10}
ight) = frac{sqrt{5}+1}{4}$

$sinleft(frac{pi}{10}
ight) = frac{sqrt{10-2sqrt{5}}}{4}$

So the coordinates of the point corresponding to $frac{pi}{10}$ on the unit circle are:

$left(frac{sqrt{5}+1}{4}, frac{sqrt{10-2sqrt{5}}}{4}
ight)$

For $frac{pi}{10}$ on the unit circle:

$cosleft(frac{pi}{10}
ight) = frac{sqrt{5}+1}{4}$

$sinleft(frac{pi}{10}
ight) = frac{sqrt{10-2sqrt{5}}}{4}$

The coordinates are:

$left(frac{sqrt{5}+1}{4}, frac{sqrt{10-2sqrt{5}}}{4}
ight)$

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