Trigonometry calculation doubt While solving trigonometry in physics especially, I often encounter values or things like $\tan \theta = 0.524$, $\tan^{-1} \frac{-3.1}{-2.4}$ ande etc...
The answer just puts the solution as $\tan^{-1} 0.524 = 27.65^o$ and $\tan^{-1} \frac{-3.1}{-2.4} = 41^0 \text{ or } 221^o$.
I want to how these values can be derived. Is there some other way or is using trigonometry tables the only way? Because I am going to attend an exam and I am not sure if trigonometry tables are provided.
 A: If you are not provided with a calculator, then in an exam like that either:


*

*You would be expected to leave any trigonometric functions in the form you find them (e.g. keep it as $\tan 27­­^\circ$); or

*You would be given a suitable approximation to use (e.g. there would be something saying "for this question, let $\tan 26^\circ = 0.51$"); or

*You would be expected to know a small set of common trig values, and they would be the ones used - the main ones are the functions of 0, 30, 45, 60 and 90 degrees.
Outside of your specific question, there are ways to find arbitrarily accurate approximations of trigonometric functions in terms of more elementary operations like addition and multiplication. One that you may eventually come across (especially in early university courses) is via Taylor series, which are (infinite) polynomials.
For example, we can write:
$$\sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \ldots$$
where $n! = 1 \times 2 \times \ldots \times n$, and $\theta$ is expressed in radians (with $\pi$ radians being equal to 180°). There are similar expressions for $\cos$ and $\tan$, as well as their inverses, along with fairly simple expressions for the error you get if you cut the calculation off after calculating some number of terms.
In fact, the trigonometry tables you've seen would have been to some extent produced via one of these formulas - they're a lot more efficient than trying to draw a perfect right-angled triangle with a specific angle and measuring the ratios of the respective sides.
