Given a unit circle centered at the origin, but flipped in a non-standard way such that the positive x-axis points downwards and the positive y-axis points to the left, find the coordinates of the point corresponding to an angle of $frac{5pi}{6}$ radian

To solve this problem, we first need to understand the transformation of the coordinate system.

In the standard unit circle, an angle of $\frac{5\pi}{6}$ radians would correspond to the point $(-\cos(\frac{\pi}{6}), \sin(\frac{\pi}{6}))$.

Therefore, in the standard unit circle, the coordinates would be:

$(-\frac{\sqrt{3}}{2}, \frac{1}{2})$

Now, since the unit circle is flipped such that the positive x-axis points downwards and the positive y-axis points to the left, we need to adjust these coordinates accordingly:

1. The x-coordinate will become the negative of the original y-coordinate.

2. The y-coordinate will become the negative of the original x-coordinate.

Thus, the transformed coordinates are:

$( -\frac{1}{2}, -\left(-\frac{\sqrt{3}}{2}\right) )$

which simplifies to:

$\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$

To find the coordinates on the flipped unit circle for an angle of $frac{5pi}{6}$, we start by determining the coordinates on the standard unit circle.

On the standard unit circle, the coordinates for $frac{5pi}{6}$ are:

$left( -cos(frac{pi}{6}), sin(frac{pi}{6})
ight)$

Which simplifies to:

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

Next, we apply the transformations needed for the flipped unit circle. In this flipped unit circle:

– The x-coordinate is derived from the negative of the y-coordinate of the standard unit circle.

– The y-coordinate is derived from the negative of the x-coordinate of the standard unit circle.

Thus, the coordinates for the flipped unit circle are:

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

First, in a standard unit circle, the coordinates for an angle of $frac{5pi}{6}$ are:

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

Given the flipped unit circle, the coordinates transform as:

$(- ext{y}, – ext{x}) = left( -frac{1}{2}, frac{sqrt{3}}{2}
ight)$

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