Determine the exact value of a trigonometric expression involving radians on the unit circle

Consider the trigonometric expression $ \cos\left(\frac{7\pi}{4}\right) + \sin\left(\frac{7\pi}{4}\right) $. Determine its exact value using the unit circle.

First, convert the given angles to radians within the unit circle:

$ \frac{7\pi}{4} $ radians is equivalent to -$ \frac{\pi}{4} $ radians (since it is in the fourth quadrant).

The coordinates of the angle -$ \frac{\pi}{4} $ are given by:

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

Thus:

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

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

Adding these values:

$ \cos\left(\frac{7\pi}{4}\right) + \sin\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2} + \left(-\frac{\sqrt{2}}{2}\right) = 0 $

Consider the trigonometric expression $ cosleft(frac{7pi}{4}
ight) + sinleft(frac{7pi}{4}
ight) $. Determine its exact value using the unit circle.

The angle $ frac{7pi}{4} $ radians is equivalent to -$ frac{pi}{4} $ radians.

The coordinates of -$ frac{pi}{4} $ are:

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

Thus:

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

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

Adding these values:

$ frac{sqrt{2}}{2} – frac{sqrt{2}}{2} = 0 $

To find the exact value of $ cosleft(frac{7pi}{4}
ight) + sinleft(frac{7pi}{4}
ight) $, use the unit circle.

The angle $ frac{7pi}{4} $ is equivalent to -$ frac{pi}{4} $ radians.

Coordinates at -$ frac{pi}{4} $ are:

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

So:

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

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

Adding these:

$ frac{sqrt{2}}{2} – frac{sqrt{2}}{2} = 0 $

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