Finding the Coordinates on the Unit Circle

Given an angle of $\frac{5\pi}{4}$ radians, find the coordinates of the point on the unit circle.

Solution:

The unit circle has a radius of 1. The coordinates for any angle $\theta$ on the unit circle can be found using the formulas $\cos(\theta)$ and $\sin(\theta)$.

Here, $\theta = \frac{5\pi}{4}$.

First, find $\cos(\frac{5\pi}{4})$:

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

Next, find $\sin(\frac{5\pi}{4})$:

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

Therefore, the coordinates are:

$(\cos(\frac{5\pi}{4}), \sin(\frac{5\pi}{4})) = (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$

Find the coordinates of the point on the unit circle corresponding to an angle of $frac{5pi}{4}$ radians.

Solution:

Using the unit circle, the coordinates for any angle $ heta$ are given by the pair $(cos( heta), sin( heta))$.

For $ heta = frac{5pi}{4}$:

Calculate $cos(frac{5pi}{4})$:

$ cos(frac{5pi}{4}) = -frac{sqrt{2}}{2} $

Calculate $sin(frac{5pi}{4})$:

$ sin(frac{5pi}{4}) = -frac{sqrt{2}}{2} $

Hence, the coordinates of the point are:

$(cos(frac{5pi}{4}), sin(frac{5pi}{4})) = (-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2})$

For an angle of $frac{5pi}{4}$ radians, find the coordinates on the unit circle.

Solution:

The coordinates are $(cos(frac{5pi}{4}), sin(frac{5pi}{4}))$.

Therefore,

$ cos(frac{5pi}{4}) = -frac{sqrt{2}}{2} $

$ sin(frac{5pi}{4}) = -frac{sqrt{2}}{2} $

The coordinates are $(-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2})$.

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