Evaluate the integral of the $ an( heta) $ function over the unit circle

To evaluate the integral of the function $ \tan(\theta) $ over the unit circle, we need to use the parametrization of the unit circle:

$ x = \cos(\theta), \quad y = \sin(\theta), \quad d\theta $

The integral over the unit circle in terms of $ \theta $ is:

$ \int_{0}^{2\pi} \tan(\theta) \cdot \frac{dy}{d\theta} \ d\theta $

Since $ \frac{dy}{d\theta} = \cos(\theta) $, we get:

$ \int_{0}^{2\pi} \tan(\theta) \cos(\theta) \ d\theta $

This integral can be simplified to:

$ \int_{0}^{2\pi} \sin(\theta) \ d\theta = 0 $

To evaluate the integral of $ an( heta) $ over the unit circle, we use:

$ x = cos( heta), quad y = sin( heta) $

Then:

$ int_{0}^{2pi} an( heta) cos( heta) d heta $

This reduces to:

$ int_{0}^{2pi} sin( heta) d heta = 0 $

We evaluate:

$ int_{0}^{2pi} sin( heta) d heta = 0 $

Start Using PopAi Today