Given a point ((x,y)) on the unit circle where the angle in radians is $ heta$, express the coordinates of the point in terms of $cos( heta)$ and $sin( heta)$, and show how these expressions can be derived from the Pythagorean identity. Additionall

The unit circle is defined by the equation:

$ x^2 + y^2 = 1 $

Since the point \((x,y)\) lies on the unit circle, we can express \(x\) and \(y\) in terms of \(\cos(\theta)\) and \(\sin(\theta)\):

$ x = \cos(\theta) $

$ y = \sin(\theta) $

These expressions satisfy the Pythagorean identity:

$ \cos^2(\theta) + \sin^2(\theta) = 1 $

Now, given \( \cos(\theta) = -\frac{1}{2} \) and \( \sin(\theta) = \frac{\sqrt{3}}{2} \), we need to find the angle \(\theta\).

The values correspond to the angle \( \theta = \frac{2\pi}{3} \) or \( \theta = \frac{4\pi}{3} \) (in the second and third quadrants respectively where cosine is negative and sine is positive).

Therefore, the angles in radians are:

$ \theta = \frac{2\pi}{3}, \frac{4\pi}{3} $

The unit circle equation is:

$ x^2 + y^2 = 1 $

For any point ((x,y)) on the unit circle, we can write:

$ x = cos( heta) $

$ y = sin( heta) $

These conform to the Pythagorean identity:

$ cos^2( heta) + sin^2( heta) = 1 $

Given ( cos( heta) = -frac{1}{2} ) and ( sin( heta) = frac{sqrt{3}}{2} ), we identify the angles by finding the intersection points of the unit circle at these coordinates. The angles are:

$ heta = frac{2pi}{3} $

$ heta = frac{4pi}{3} $

Given the unit circle:

$ x^2 + y^2 = 1 $

The coordinates of a point are:

$ x = cos( heta) $

$ y = sin( heta) $

Given ( cos( heta) = -frac{1}{2} ) and ( sin( heta) = frac{sqrt{3}}{2} ), the angles are:

$ heta = frac{2pi}{3}, frac{4pi}{3} $

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