Determine the value of $ heta $ for which the point $ (cos( heta), sin( heta)) $ on the unit circle forms a right-angled triangle with the origin and the point $ (1, 0) $

Given the points $ (\cos(\theta), \sin(\theta))$, the origin $(0, 0)$, and $(1, 0)$, we need to find $ \theta $ such that they form a right-angled triangle.

The distance between $(\cos(\theta), \sin(\theta))$ and $(1, 0)$ is:

$ d = \sqrt{(\cos(\theta) – 1)^2 + \sin^2(\theta)} $

Since $(\cos(\theta), \sin(\theta))$ lies on the unit circle, we use the Pythagorean identity:

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

Thus the distance simplifies to:

$ d = \sqrt{1 – 2\cos(\theta) + 1} = \sqrt{2 – 2\cos(\theta)} = \sqrt{2(1 – \cos(\theta))} $

For the triangle to be right-angled, $\cos(\theta) = \frac{1}{2}$:

$ \cos(\theta) = \frac{1}{2} \implies \theta = \frac{\pi}{3} \text{ or } \theta = -\frac{\pi}{3} $

For the points $(cos( heta), sin( heta))$ and $(1, 0)$ to form a right-angled triangle with the origin, the angle $ heta$ must satisfy:

$ cos( heta) = frac{1}{2} $

Therefore, the values of $ heta$ are:

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

To form a right-angled triangle, $cos( heta) = frac{1}{2}$ is required:

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

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