TITLE: Is there a way to get trig functions without a calculator?
QUESTION [58 upvotes]: In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?
Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.
Or, if you could redirect me to a place that explains how to do it, please do so.
My dad said there isn't, but I just had to make sure.
Thanks.

REPLY [2 votes]: Long before there were power series, in the second century A.D., Ptolemy, a man who wrote in Greek and probably lived in Alexandria, created a table of values of what amounts to the sine function.
See this page.

Chapter 10 of Book I of the ''Almagest'' presents geometric theorems used for computing chords.  Ptolemy used geometric reasoning based on Proposition 10 of Book XIII of Euclid's Elements to find the chords of $72^\circ$ and $36^\circ.$  That Proposition states that if an equilateral pentagon is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle. He used Ptolemy's theorem on quadrilaterals inscribed in a circle to derive formulas for the chord of a half-arc, the chord of the sum of two arcs, and the chord of a difference of two arcs.  The theorem states that for a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the two pairs of lengths of opposite sides.  The derivations of trigonometric identities rely on a cyclic quadrilateral in which one side is a diameter of the circle. To find the chords of arcs of $1^\circ$ and $\left(\tfrac 1 2\right)^\circ$ he used approximations based on Aristarchus's inequality.  The inequality states that for arcs $\alpha$ and $\beta,$ if $0<\beta<\alpha<90^\circ,$ then $$ \frac{\sin \alpha}{\sin \beta} < \frac\alpha\beta < \frac{\tan\alpha}{\tan\beta}.$$ Ptolemy showed that for arcs of $1^\circ$ and $\left(\tfrac 1 2 \right)^\circ,$ the approximations correctly give the first two sexigesimal places after the integer part.