# How to solve trigonometric equations without a calculator? [duplicate]

How does one solve for sine, cosine, or tangent (and inverse sine, cosine, tangent) without the use of a calculatour? For example: cosine of 131 degrees — how must one calculate this?

## marked as duplicate by Claude Leibovici, Crostul, TastyRomeo, Lucian, kingW3Jan 13 '17 at 13:37

• Taylor series would be a good start – Oiler Jan 13 '17 at 4:48
• To approximate, use well-known values and continually average them. For example, cos(135) ~ -0.7071 and cos(120) = -0.5. Averaging these gives -0.6035. 131 is closer to 135, so average this new number with cos(135) again to get -0.6553, and the actual value is cos(131) = 0.65605... – infinitylord Jan 13 '17 at 5:34
• It is not clear whether 1) you desire to know the method used by calculators or 2) you need an efficient method for you, for hand calculation? One thing is certain : using degrees complicates the task. It is much preferable to work with radian units – Jean Marie Jan 13 '17 at 5:46

Let's figure out what happen to the earlier people who need to compute the trigonometric function out of paper and pencil or perhaps an abacus.

As a starter, we can always compute the square root using the Newton's algorithm.

Next suppose we wanted to compute $\cos(a + b)$ such that we know $\sin(a)$, $\cos(a)$, $\sin(b)$, $\cos(b)$, we could do $\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$

Also, we can compute $\cos(x) = \sqrt{\frac{1 + \cos(2x)}{2}}$

Of course, $\sin(x) = \sqrt{1 - \cos^2(x)}$, so is also computable.

So together, given $\sin(a)$, $\cos(a)$, $\sin(b)$, $\cos(b)$, we can compute $\sin(\frac{a + b}{2})$ and $\cos(\frac{a + b}{2})$.

So given a handful of know values, we can approximate the trigonometric function by getting closer and closer to the value you want, this is basically bisection.

Of course, that is a lot of work, that is precisely why they have tables.

In a calculator, this is typically implemented using the CORDIC algorithm

https://en.wikipedia.org/wiki/CORDIC

The basic principle is similar there.