Given a point P on the unit circle at an angle $ heta $ (in radians) from the positive x-axis, find the coordinates of P if $ heta $ is transformed by the function $ f( heta) = 2 heta + frac{pi}{4} $. Also, identify the new x and y coordinates aft

Given the initial angle $ \theta $, the coordinates of the point $ P $ are:

$ ( \cos \theta, \sin \theta ) $

With the transformation $ f(\theta) = 2\theta + \frac{\pi}{4} $, let the new angle be $ \theta’ = 2\theta + \frac{\pi}{4} $. The new coordinates of the point $ P’ $ are:

$ ( \cos(2\theta + \frac{\pi}{4}), \sin(2\theta + \frac{\pi}{4}) ) $

For example, if $ \theta = \frac{\pi}{6} $, then:

$ \theta’ = 2\times \frac{\pi}{6} + \frac{\pi}{4} = \frac{\pi}{3} + \frac{\pi}{4} = \frac{7\pi}{12} $

Thus, the new coordinates are:

$ P’ ( \cos \frac{7\pi}{12}, \sin \frac{7\pi}{12} ) $

Initially, we have the coordinates $ ( cos heta, sin heta ) $. After applying $ f( heta) = 2 heta + frac{pi}{4} $, the angle $ heta’ = 2 heta + frac{pi}{4} $. The new coordinates are:

$ ( cos(2 heta + frac{pi}{4}), sin(2 heta + frac{pi}{4}) ) $

Assuming $ heta = frac{pi}{4} $, then:

$ heta’ = 2 imes frac{pi}{4} + frac{pi}{4} = frac{3pi}{4} $

So, the new coordinates are:

$ P’ ( cos frac{3pi}{4}, sin frac{3pi}{4} ) = ( -frac{sqrt{2}}{2}, frac{sqrt{2}}{2} ) $

Given $ ( cos heta, sin heta ) $ and $ f( heta) = 2 heta + frac{pi}{4} $, the new coordinates are:

$ ( cos(2 heta + frac{pi}{4}), sin(2 heta + frac{pi}{4}) ) $

If $ heta = 0 $, then:

$ heta’ = frac{pi}{4} $

So, the new coordinates are:

$ P’ ( cos frac{pi}{4}, sin frac{pi}{4} ) = ( frac{sqrt{2}}{2}, frac{sqrt{2}}{2} ) $

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