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Arctan Formula

In mathematics subject every function has an inverse. Similarly trigonometric function also comprise inverse. In trigonometry arctan is the inverse of the tangent function, and is used to compute the angle measure from the tangent ratio (tan = opposite/adjacent) of a right triangle. Arctan can be calculated in terms of degrees and as well as radians.

$\large \arctan (x)=2\arctan \left ( \frac{x}{1+\sqrt{1+x^{2}}} \right )$

$\large \arctan (x)=\int_{0}^{x}\frac{1}{z^{2}+1}dz\;;\;\left | x \right |\leq 1$

$\large \int \arctan (z)dz=z\arctan (z)-\frac{1}{2}\ln (1+z^{2})+C$

Arctangent formulas for $\pi$

$\large \frac{\pi }{4}=4\arctan \frac{1}{5}-\arctan \frac{1}{239}$

$\large \frac{\pi }{4}=\arctan\frac{1}{2}+\arctan \frac{1}{3}$

$\large \frac{\pi }{4}=2\arctan\frac{1}{2}-\arctan \frac{1}{7}$

$\large \frac{\pi }{4}=2\arctan\frac{1}{3}+\arctan \frac{1}{7}$

$\large \frac{\pi }{4}=8\arctan\frac{1}{10}-4\arctan \frac{1}{515}-\arctan \frac{1}{239}$

$\large \frac{\pi }{4}=3\arctan\frac{1}{4}-\arctan \frac{1}{20}-\arctan \frac{1}{1985}$

$\large \frac{\pi }{4}=24\arctan\frac{1}{8}-8\arctan \frac{1}{57}-4\arctan \frac{1}{239}$

Solved Examples

Question 1: Evaluate tan-1(1.732)?

Solution:

Given value is, tan-1(1.732)

From this given quantity, 1.732 can be written as a function of tan.

So, 1.732 = tan(60)

Therefore, tan-1(1.732) = tan-1 (tan(60)) = 60°

60° = 60 $\times$ $\frac{\pi}{180}$ = 1.047 radians.