EXAMPLES LET A = **ARCTAN**(-2) LET A = **ARCTAN**(A1) LET X2 = **ARCTAN**(X1-4) DEFAULT None SYNONYMS None RELATED COMMANDS ARCCOS = Compute arccosine. ARCCOSH = Compute hyperbolic arccosine.

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NOTES Edited by William Adkins A Sequence of Polynomials for Approximating Arctangent Herbert A. Medina 1. INTRODUCTION. The Taylorseries **arctan** x = x − x 3 3 + x 5 5 −+···= ∞ * k =0 ( −1 ) k 2 k +1 x 2 k +1 was discovered by the Scotsman James Gregory in 1671[ 1 , chap. 12].

In general the larger N becomes the more rapidly the infinite series for **arctan**(z) will converge. Thus the series for (π/8) =**arctan**{ 1/[1+sqrt(2)]} reads -

Since θ = **arctan** ( B/x ) - **arctan** ( A/x ) , d ...

**arctan**.DVI. A Taylorseriesfor the functionarctan The integral Ifweinverty=**arctan**(x) to obtainx=tany, then, by differentiating with respect toy, we finddx/dy=sec 2 y=1+tan 2 y=1+x 2.

Since the area of a sector of a circle of radius r and angle (in radians) is the total area of this shaded region is **Arctan** a . We shall show that these two shaded regions have equal areas.

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MIT OpenCourseWare http://ocw.mit.edu 18.01SC Single Variable Calculus Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

FOUR QUADRANT APPROXIMATIONS The **arctan** algorithms presented thus far are applicable for angles in the range of − π/ 4to π/ 4rad. Here we extend the angular range to − π to π radians.