Find the coordinates of the point where the terminal side of an angle in standard position intersects the unit circle if the angle is given by ( heta = frac{7pi}{4} )

The unit circle has a radius of 1 and is centered at the origin. The coordinates (x, y) on the unit circle for an angle \( \theta \) are given by:

$ (x, y) = (\cos(\theta), \sin(\theta)) $

For \( \theta = \frac{7\pi}{4} \):

$ \cos \left( \frac{7\pi}{4} \right) = \cos \left( 2\pi – \frac{\pi}{4} \right) = \cos \left( \frac{-\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $

$ \sin \left( \frac{7\pi}{4} \right) = \sin \left( 2\pi – \frac{\pi}{4} \right) = \sin \left( \frac{-\pi}{4} \right) = -\sin \left( \frac{\pi}{4} \right) = -\frac{\sqrt{2}}{2} $

Therefore, the coordinates are:

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

For an angle ( heta = frac{7pi}{4} ), find the coordinates on the unit circle:

$ (x, y) = (cos( heta), sin( heta)) $

Using ( heta = frac{7pi}{4} ):

$ cos left( frac{7pi}{4}
ight) = frac{sqrt{2}}{2} $

$ sin left( frac{7pi}{4}
ight) = -frac{sqrt{2}}{2} $

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

Coordinates for ( heta = frac{7pi}{4} ):

$ left( cos(frac{7pi}{4}), sin(frac{7pi}{4})
ight) = left( frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}
ight) $

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