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Why we take unit circle in trigonometry? If we take circle of radius 2 then how does it mess up with values of trig functions at given angles?

thanks

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    For the sake of simplicity. (The figures would be similar and the ratios would be the same.) – 2017-01-26

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There is a deep connection with the circumference of the circle then. The angle in the unit circle (measured in radians) gives the corresponding part of the circumference of the circle.

Further, we can define cosine and sine using the circle as the orthogonal projections on the x-axis and y-axis. We have the property that cosine and sine values are always between $-1$ and $1$. With some accurate drawing, one does not even have to remember things like $\cos(\frac{\pi}{3}) = \frac{1}{2}$.

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    what is orthogonal projection – 2017-01-26
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    Take a look at https://en.wikipedia.org/wiki/Projection_(linear_algebra)#Orthogonal_projections – 2017-01-26
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If we take $r=1$ we can write directly $x=cos(\alpha)$ and $y=sin(\alpha)$ what is simpler than adding to formulas also $r$.