Given a point on the unit circle at an angle of $frac{3pi}{4}$ radians, find the coordinates of this point and verify the trigonometric identities for sine and cosine at this angle.

To solve this problem, we first need to understand the unit circle and the angle $\frac{3\pi}{4}$ radians.

On the unit circle, the angle $\frac{3\pi}{4}$ is located in the second quadrant where sine is positive and cosine is negative. The reference angle for $\frac{3\pi}{4}$ radians is $\frac{\pi}{4}$ radians.

For the reference angle $\frac{\pi}{4}$, the sine and cosine values are both equal to $\frac{\sqrt{2}}{2}$.

Therefore, at $\frac{3\pi}{4}$ radians, the sine is positive, and the cosine is negative:

$\sin\left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2}$

$\cos\left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2}$

So, the coordinates of the point at $\frac{3\pi}{4}$ radians on the unit circle are:

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

To verify the trigonometric identities, we can check:

$\sin^2\left( \frac{3\pi}{4} \right) + \cos^2\left( \frac{3\pi}{4} \right) = \left( \frac{\sqrt{2}}{2} \right)^2 + \left( -\frac{\sqrt{2}}{2} \right)^2 = \frac{1}{2} + \frac{1}{2} = 1$

Thus, the identities are verified.

The unit circle allows us to determine the coordinates of a point at any given angle. For an angle of $frac{3pi}{4}$ radians, we are in the second quadrant.

In this quadrant, the sine value is positive, and the cosine value is negative. The reference angle here is $frac{pi}{4}$ radians, which has known sine and cosine values of $frac{sqrt{2}}{2}$.

The coordinates of the point at $frac{3pi}{4}$ radians can be calculated as:

$left( cosleft( frac{3pi}{4}
ight), sinleft( frac{3pi}{4}
ight)
ight) = left( -frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$

To confirm this, we use the Pythagorean identity:

$sin^2left( frac{3pi}{4}
ight) + cos^2left( frac{3pi}{4}
ight) = 1$

Substituting the values:

$left( frac{sqrt{2}}{2}
ight)^2 + left( -frac{sqrt{2}}{2}
ight)^2 = frac{1}{2} + frac{1}{2} = 1$

This confirms the trigonometric identities.

At $frac{3pi}{4}$ radians, which is in the second quadrant:

$sinleft( frac{3pi}{4}
ight) = frac{sqrt{2}}{2}$

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