Find the sine, cosine, and tangent of the angle formed by a point on the unit circle at $frac{5pi}{4}$ radians.

To find the sine, cosine, and tangent of the angle $\frac{5\pi}{4}$ radians, we need to locate the point on the unit circle corresponding to this angle.

First, let’s convert $\frac{5\pi}{4}$ radians to degrees. We know that $\pi$ radians is equivalent to $180^\circ$, so:

$\frac{5\pi}{4} \times \frac{180^\circ}{\pi} = 225^\circ$

The angle $225^\circ$ lies in the third quadrant, where both sine and cosine are negative.

The reference angle for $225^\circ$ is:

$225^\circ – 180^\circ = 45^\circ$

For $45^\circ$, the sine and cosine values are $\frac{\sqrt{2}}{2}$. Since $225^\circ$ is in the third quadrant, we have:

$\sin(225^\circ) = -\frac{\sqrt{2}}{2}$

$\cos(225^\circ) = -\frac{\sqrt{2}}{2}$

The tangent is given by the ratio of sine to cosine:

$\tan(225^\circ) = \frac{\sin(225^\circ)}{\cos(225^\circ)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$

To determine the sine, cosine, and tangent of the angle $frac{5pi}{4}$ radians, identify the position on the unit circle.

Converting $frac{5pi}{4}$ to degrees:

$frac{5pi}{4} imes frac{180^circ}{pi} = 225^circ$

Since $225^circ$ is in the third quadrant, both sine and cosine are negative.

The reference angle for $225^circ$ is:

$225^circ – 180^circ = 45^circ$

Using $frac{sqrt{2}}{2}$ for $45^circ$, we get:

$sin(225^circ) = -frac{sqrt{2}}{2}$

$cos(225^circ) = -frac{sqrt{2}}{2}$

The tangent is:

$ an(225^circ) = frac{sin(225^circ)}{cos(225^circ)} = 1$

For $frac{5pi}{4}$ radians:

$frac{5pi}{4} imes frac{180^circ}{pi} = 225^circ$

In the third quadrant, reference angle $225^circ – 180^circ = 45^circ$:

$sin(225^circ) = -frac{sqrt{2}}{2}$

$cos(225^circ) = -frac{sqrt{2}}{2}$

$ an(225^circ) = 1$

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