Determine the values of $ heta $ where $ sin( heta) $ and $ cos( heta) $ are equal in the flipped unit circle

To determine the values of $ \theta $ where $ \sin(\theta) $ and $ \cos(\theta) $ are equal in the flipped unit circle, we start by setting up the equation:

$ \sin(\theta) = \cos(\theta) $

Dividing both sides by $ \cos(\theta) $, we get:

$ \tan(\theta) = 1 $

In the standard unit circle, $ \tan(\theta) = 1 $ when $ \theta = \frac{\pi}{4} + k\pi $, where $ k $ is an integer. However, since this is a flipped unit circle, we need to consider transformations:

$ \theta = -\left(\frac{\pi}{4} + k\pi \right) $

Hence, the values of $ \theta $ are given by:

$ \theta = -\frac{\pi}{4} – k\pi $

To find the values of $ heta $ where $ sin( heta) $ and $ cos( heta) $ are equal on a flipped unit circle, start with:

$ sin( heta) = cos( heta) $

Now:

$ an( heta) = 1 $

In the standard unit circle:

$ heta = frac{pi}{4} + kpi $

In the flipped unit circle, transformations give:

$ heta = -left(frac{pi}{4} + kpi
ight) $

To find where $ sin( heta) = cos( heta) $ in the flipped unit circle, solve:

$ an( heta) = 1 $

In the flipped unit circle:

$ heta = -left(frac{pi}{4} + kpi
ight) $

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