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You are here: Home → Articles → How does a calculator...
How does the calculator find values of sine (or cosine or tangent)?
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:
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).
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.
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