$ ext{Determine the coordinates of a point on a unit circle with a given angle}$

Let’s find the coordinates of a point on the unit circle corresponding to an angle of $\frac{5\pi}{4}$ radians.

The unit circle equation is given by:

$x^2 + y^2 = 1$

For an angle $\theta$, the coordinates $(x, y)$ are given by:

$x = \cos(\theta)$

$y = \sin(\theta)$

Substituting $\theta = \frac{5\pi}{4}$:

$x = \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

$y = \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

Hence, the coordinates of the point are:

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

To determine the coordinates of a point on the unit circle at the angle of $frac{7pi}{6}$ radians, we use the unit circle properties.

Recall that:

$x = cos( heta)$

$y = sin( heta)$

For $ heta = frac{7pi}{6}$:

$x = cosleft(frac{7pi}{6}
ight) = -frac{sqrt{3}}{2}$

$y = sinleft(frac{7pi}{6}
ight) = -frac{1}{2}$

Thus, the point’s coordinates are:

$oxed{left(-frac{sqrt{3}}{2}, -frac{1}{2}
ight)}$

For the angle $frac{11pi}{6}$ radians on a unit circle:

$x = cosleft(frac{11pi}{6}
ight) = frac{sqrt{3}}{2}$

$y = sinleft(frac{11pi}{6}
ight) = -frac{1}{2}$

Coordinates:

$oxed{left(frac{sqrt{3}}{2}, -frac{1}{2}
ight)}$

Start Using PopAi Today