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How does the calculator find values of sine (or cosine or tangent)?

Exactly what happens when i type the sine (or cos or tan etc for that matter) of an angle into my calculator? I type it in and it magically gives me an answer, a number that is essentially unrelated to the angle I inputed. Is the calculator just reading off of a list created from people who used rulers to physically measure the distance on a graph or is there a mathematical function that defines it?

A calculator or computer program is not reading off of a list, but is using an algorithm that gives an approximate value of sine of a given angle. There are several such algorithms. Many times they only use the four basic operations (+, -, x, /) to find the sine or cosine or tangent of a given angle.

One way is to take a certain amount of terms (the more terms, the more accurate the approximation) from the Taylor series for sine:

sin x = x - x3/3! + x5/5! - x7/7! + ...,

where x is in radians. For example, to find out sine 23, first convert 23 to radians by dividing it by 180 and then multiplying by p. We get 23/180 p = 0.401425727958696 0.4014257. Then use the above formula to get the value of sin 0.4014257:

sin 0.4014257 = 0.4014257 - 0.40142573/3! + 0.40142575/5! - 0.40142577/7!
= 0.4014257 - 0.0107811296737492 + 0.000086864959350 - 0.000000333277256
= 0.390731102008345

Calculator would give directly Sin 23 = 0.390731128489274 so the formula hit it right for the first seven decimals (and there even exists and error estimate that tells you how big an error maximum you make when you use a certain number of terms from the Taylor series).
See how closely the polynomial formed from the first four terms of the Taylor series approximates the sine curve near origin.

In the picture they look identical but we know they are not exactly the same, just very very close - even the above calculation shows that.
graph of sine and its Taylor polynomial

Calculators often use the CORDIC algorithm to find values of trigonometric functions, which is based on thinking of the angle as the phase of a complex number in the complex plane, and then rotating the complex number by multiplying it by a succession of constant values.

In fact, the calculator or computer program uses some kind of algorithm based on the basic operations to calculate not only trigonometric values, but also square roots, values of hyperbolic functions and others. The branch of mathematics called numerical methods studies and develops these algorithms. It is an interesting field of mathematics, and one that surely has lots of applications. Of course if you study computer science you will encounter many of them.

See more information:

Calculating Sine Without Using the Sine Key or a Table
The Magic Calculator and the Sine Addition Formula
Sine Waves - A collection of different ways to generate sine waves in software.

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