Determine the sine and cosine of the angle $frac{pi}{4}$ on the unit circle.

To find the sine and cosine of the angle $\frac{\pi}{4}$ on the unit circle, we use the definitions of sine and cosine for the unit circle.

For an angle $\theta$ in the unit circle, $\cos(\theta)$ is the x-coordinate and $\sin(\theta)$ is the y-coordinate of the corresponding point.

At $\theta = \frac{\pi}{4}$, the coordinates are known to be $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Thus,

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

and

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

The unit circle allows us to determine the sine and cosine for the angle $frac{pi}{4}$. In the unit circle, the coordinates of the point where the terminal side of the angle intersects the circle give the cosine and sine of the angle.

Specifically,

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

and

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

The coordinates of the point are $left(frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$, confirming that:

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

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

The angle $frac{pi}{4}$ on the unit circle has coordinates:

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

Therefore,

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

and

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

Start Using PopAi Today