Guest Post: Calculating Trignometric Values

Many students who start to learn Trigonometry often ask themselves how do we come to know that

\sin 30^\circ = \frac{1}{2} or \cos 45 ^\circ = \frac{\sqrt{2}}{2}

or for that matter any other trigonometric value?

Most of us would say use a trigonometric table or use a scientific calculator and you get the value. That’s okay, but the question still remains unanswered. How does a calculator come to know that \tan 15^\circ = 0.26794919... or how did the mathematicians create the entire trig tables when calculators were not invented? There should be some formula that tells us as to how the values are calculated. More importantly, can I, using a standard calculator, find the approximate value of let’s say \sin 50.5^\circ? Yes, there is a simple formula to find the value of sine of any acute angle. Though the formula does not give accurate results, it comes handy to know the value of \sin \theta  where 0^\circ \leq \theta \leq 90^\circ.

 \sin x ^\circ = \frac{4x(180 - x)}{40500 - x(180 - x)}.

This rational approximate formula was discovered by Bhaskara I of India in the seventh century. This simple formula enables us to calculate the sine of any given acute angle (any even obtuse angle) with a maximum absolute error of 0.00163.  » Read more