In trigonometry, the tangent is defined as the ratio of the opposite side to the adjacent side of a specific angle of a right-angled triangle whereas arctan is the inverse of the tangent function and is used for finding the angle. Arctan formula is used for this purpose.

Arctan is also known as the tan-1. The arctan formula is explained using solved examples below.

What is the arctan formula?

Basic arctan formula is expressed as:

  • θ =  arctan(opposite ÷ adjacent)

Other arctan formulas are:

  • arctan(x) = 2arctan \(\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)\)
  • arctan(x) = \(\int_{0}^{x} \frac{1}{z^{2}+1} d z ;|x| \leq 1 \)
  • ∫arctan(z) dz = z arctan(z) - \(\frac{1}{2} \ln \left(1+z^{2}\right)\) + C  

Arctangent formulas for π

  • π/4 = 4 arctan(1/5) - arctan(1/239)
  • π/4 = arctan(1/2) + arctan(1/3)
  • π/4 = 2 arctan(1/2) - arctan(1/7)
  • π/4 = 2 arctan(1/3) + arctan(1/7)
  • π/4 = 8 arctan(1/10) - 4 arctan(1/515) - arctan(1/239)
  • π/4 = 3 arctan(1/4) + arctan(1/20) + arctan(1/1985)
  • π/4 = 24 arctan(1/8) + 8 arctan(1/57) + 4 arctan(1/239)

Solved Examples Using arctan formula

Example 1

In the right-angled triangle ABC, the base of the triangle is 23 and the height is 15. Find the base angle.

Solution

To find: base angle

Using arctan formula

θ =  arctan(opposite ÷ adjacent)

θ = arctan(15 ÷ 23) = arctan(0.65)

θ = 33 degrees or 33o.

Answer: The angle is 33o.

Example 2

In the right-angled triangle ABC, if the base of the triangle is 2 units and the height of the triangle is 3 units. Find the base angle.

Solution

To find: base angle

Using arctan formula

θ =  arctan(opposite ÷ adjacent)

θ = arctan(3 ÷ 2) = arctan1.5

θ = 56o

Answer: The angle is 56o.