How calculators do trigonometry I need to use a calculator to find trigonometric ratios like $\sin(41),\cos(32)$.

How do calculators work for trigonometry? Also, as calculators are invented by the human mind can we do it mentally? Is there any binary trick or something like that?
Here is a page on the TI-Knowledge base asserting that TI calculators use a well established algorithm called the CORDIC algorithm. This algorithm is well described on Wikipedia and in a great paper from the College Math Journal.

To be clear, while Taylor series (referred to in all the previous answers) are very important in numerical analysis they are not the tool of choice for computation of trig functions at arbitrary values.
Basically, calculators are able to perform additions, subtractions, multiplications and, somewhat less easily, divisions. They are also able to store precomputed constants, and perform table lookups (select a constant among a series, based on an integer index).

Equipped with these primitive operations, calculators can essentially evaluate piecewise polynomials or rational fractions, i.e. functions defined by polynomials or the ratio of polynomials in some intervals.

There are several approximation techniques available, such as the Taylor development, Lagrangian interpolation, Padé approximants, splines... Such techniques work well for smooth functions, including the trigonometric ones. Among them, Chebychev interpolation is one of the most interesting, as it minimizes the error between the true function and the polynomial, for a given number of terms.

For instance,

$$\cos\left(\frac\pi2x\right)\approx0.9994032-1.2227967x^2+0.2239903x^4,x\in[-1,1]$$



In the case of trigonometric functions, an important ad-hoc method, called CORDIC, can be used. In a nutshell, it relies on the addition formulas

$$\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b),\\
\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b).$$

Then by a clever decomposition of the argument $x$ as a sum of decreasing terms, the functions can be computed very cheaply.