Calculate the cosine and sine of a (45^circ) angle using the unit circle.

To find the cosine and sine of a \(45^\circ\) angle, we use the unit circle, where the radius is 1.

In the unit circle, a \(45^\circ\) angle corresponds to \(\frac{\pi}{4}\) radians.

The coordinates of this point are \(\left( \cos \frac{\pi}{4}, \sin \frac{\pi}{4} \right)\).

For \(\frac{\pi}{4}\) radians:

$ \cos \left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $

$ \sin \left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $

Therefore, the cosine and sine of a 45-degree angle are both \(\frac{\sqrt{2}}{2}\).

To find the cosine and sine of (45^circ), use the unit circle where the coordinates of the point corresponding to (45^circ) are (left( cos 45^circ, sin 45^circ
ight)).

In the unit circle, the angle (45^circ) can be written as (frac{pi}{4}) radians.

We know that:

$ cos frac{pi}{4} = frac{1}{sqrt{2}} = frac{sqrt{2}}{2} $

$ sin frac{pi}{4} = frac{1}{sqrt{2}} = frac{sqrt{2}}{2} $

Thus, the cosine and sine values for a (45^circ) angle are respectively (frac{sqrt{2}}{2}) and (frac{sqrt{2}}{2}).

Using the unit circle, find ( cos ) and ( sin ) for a ( 45^circ ) angle:

$ cos 45^circ = frac{sqrt{2}}{2} $

$ sin 45^circ = frac{sqrt{2}}{2} $

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