# Arctan Formula

## Arctan Formula

Trigonometry is the science of evaluating and demonstrating the sides and angles of a right-angled triangle. Trigonometric operations are performed using sides, angles, and trigonometric ratios. These trigonometric ratios are the values of trigonometric functions calculated from the ratios of the triangle’s sides and angles.

Trigonometry has fundamental trigonometric functions, each with its standard trigonometric ratio value under different angles. The fundamental functions are sine, cosine, tangent, cotangent, cosecant, and secant. The inverse of several trigonometric functions, such as arcsin, arccos, arctan, arccot, arcsec, and arccosec, is denoted by the prefix ‘arc-‘.

Every function in mathematics has an inverse. Similarly, the trigonometric function includes an inverse. In trigonometry, the Arctan Formula is the inverse of the tangent function and is used to determine the angle measure from a right triangle’s tangent ratio (tan = opposite/adjacent). The Arctan Formula can be expressed in terms of degrees and radians.

θ =arctan(perpendicular/base)

arctan(-x)=-arctan(x) for all x∈ R

tan(arctan x)=x, for all real numbers

arctan(1/x)=π/2 – arctan(x) = arccot(x) ; if x>0

(OR)

arctan(1/x)=-π/2 – arctan(x) = arccot(x) -π ; if x<0

sin(arctan x)= x/ √(1+x2)

cos(arctan x)=1/ √(1+x2)

arctan(x)=2arctan(\frac{x}{1+\sqrt(1+x^2)})

arctan(x)=\int^x_0\frac{1}{z^2+1}dz

#### Arctangent formulas for π

As previously stated, the fundamental of the Arctan Formula is arctan (Perpendicular/Base) =θ, where θ is the angle between the hypotenuse and the base of a right-angled triangle. Researchers use the  Arctan Formula to get the value of an angle θ in degrees or radians. One can also write the Arctan Formula as θ = tan-1[Perpendicular / Base].

4= 4arctan 15 – arctan 1239

4= arctan 12 + arctan 13

4= 2arctan 12 – arctan 17

4= 2arctan 12 + arctan 17

4= 8arctan 110 -4arctan 1515-arctan 1239

4= 3arctan 14 +4arctan 120+arctan 11985

4= 24arctan 18 +8arctan 157+4arctan 1239

### Solved Example

Evaluate: tan-1(1.732)

The 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×π180