Find the coordinates of a point on the negative unit circle given a specific angle $ heta $

To find the coordinates of a point on the negative unit circle given a specific angle $ \theta $, we use the equation of the unit circle:

$ x^2 + y^2 = 1 $

The coordinates can be found using parametric equations:

$ x = -\cos(\theta) $

$ y = -\sin(\theta) $

For example, if $ \theta = \frac{\pi}{4} $, the coordinates are:

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

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

Thus, the coordinates at $ \theta = \frac{\pi}{4} $ are:

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

Using the unit circle equation $ x^2 + y^2 = 1 $ and parametric equations:

$ x = -cos( heta) $

$ y = -sin( heta) $

For $ heta = frac{pi}{4} $:

$ x = -frac{sqrt{2}}{2} $

$ y = -frac{sqrt{2}}{2} $

Coordinates: $ left( -frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}
ight) $

Using the parametric equations for the unit circle:

$ x = -cos( heta) $

$ y = -sin( heta) $

At $ heta = frac{pi}{4} $:

Coordinates: $ left( -frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}
ight) $

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