VIDEO: Calculate Arctan without a calculator

The approximation formula for arctan

• The arctangent is the inverse function of the normal tangent function. With arctan (x) you calculate the corresponding arc y for a tangent value x (from which you can then determine the angle).
• For practically all angle functions and their inverse functions there are approximation formulas with which they can be calculated. The formulas are based on this Functions using a power series that contain the argument x of the function in increasing powers.
• For the arctangent, the inverse function of the tangent, the following development applies: arctan (x) = x - x³/ 3 + x⁵/5... However, the amount of the argument x must not be greater than 1. The approximation is more accurate the more powers you take into account when calculating. In most cases in which one works by hand, i.e. without a calculator, the first two terms are quite sufficient.

By the way: For very small values ​​of x the following applies: actan (x) ≈ x, in many cases also a sufficient approximation. The power series can also be used if one

Equations should solve with trigonometric functions, in which "x" also occurs in simple form.

Calculate without a calculator - an example

In this example x = 0.5 should be chosen. You are looking for the corresponding sheet or the angle.

1. You apply the approximation formula mentioned above, for example the first two terms of the power series expansion. The following applies: arctan (0.5) = 0.5 - (0.5)³/3 = 0,5 - 0,042 = 0,458.
2. For this arc x you calculate the angle α = x _* 180°/ π = 0,458 _* 180° / π = 26,24°.

You can now approximate with the calculator check again. The following applies: tan (26.24 °) = 0.492 and arctan (0.5) = 26.57 °. Please note that when calculating the angle, the angle was rounded twice - so the result is when only two are taken into account Potencies quite acceptable. If you work without a calculator, it can be difficult to carry further powers with you by means of written exponentiation.