Approximation Formula

a mathematical formula obtained from an expression of the form f(x) = f*(x) + ∊(x), where ∊(x) is regarded as the error and after evaluation is dropped. Thus, an approximation formula has the form f (x) ≈ f*(x). For example, the approximation formula (1 + x)² ≈ 1 + 2x can be obtained from the exact formula for (1 + x)² for small ǀxǀ; when the formula is used in computation, the result is accurate to two, three, or four decimal places if ǀxǀ is less than 0.0707 …, 0.0223 …, or 0.00707 …, respectively. The closer x approaches 0, the more accurate is the result given by the formula. But this is not always the case. For example, the accuracy of the approximation formula

increases as x approaches π/2.

Table 1 presents some commonly used approximation formulas together with an indication of the number that ǀxǀ must not exceed in order that the formula be accurate to k decimal places.

Table 1
Formula	k = 2	k = 3	k = 4
(1 + x)3 ≈ 1 + 3x	0.04	0.012	0.004
	0.06	0.022	0.007
	0.19	0.062	0.020
	0.20	0.065	0.021
sin x ≈ x	0.31 (17°48ʹ)	0.144 (8°15ʹ)	0.067 (3°50ʹ)
cos x≈ 1	0.10 (5°43ʹ)	0.031 (1°48ʹ)	0.010 (0°34ʹ)
tan x≈ x	0.25 (14°8ʹ)	0.112 (6°25ʹ)	0.053 (3°2ʹ)
log (1 + x) ≈ 0.4343x	0.14	0.47	0.015
10x ≈ 1 + 2.303x	0.04	0.014	0.004
	0.25	0.119	0.055

Approximation formulas are frequently obtained by means of the expansion of functions in series, such as a Taylor series. In order to use an approximation formula with confidence, we must have an estimate of the difference between the exact and the approximate expressions for the function. If we know, for example, that the difference between sin x and the binomial x – x³/6 does not exceed x⁵/120 in absolute value, we can easily see that the approximation formula sin x ≈ x – x³/6 gives the value of sin x accurate to two, three, or four decimal places if x is less than 0.89 (51°), 0.55 (32°), or 0.34 (20°), respectively.