Arctan Formula

What is Arctan?

In trigonometry mathematics, every function has an inverse and arctan is the inverse of the tangent function. Arctan is also referred to as the tan-1. Arctan x is used to find the angle. The tangent on the other hand is described as the ratio of the opposite side to the adjacent side of a particular angle of a right-angled triangle. Arctan formula is used to identify an angle.


What is the Arctan Formula?

A fundamental arctan formula is written as:

  • θ =  arctan(opposite ÷ adjacent)

Other arctan formulas are as given below:

  • arctan(x) = 2arctan (x/1+√1+x2)

  • arctan(x) = ∫x0 1/z2+1dz;|x|≤1

  • ∫arctan(z) dz = z arctan(z) – 1/2 ln(1+z2) + C  

Arctangent formulas for π are as given below:

  • 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

The arctan formula can be thoroughly understood for use and application referring to solved examples below.


Example:

In the right-angled triangle PQR, the base of which measures 17 cm and the height is 9cm. Determine the base angle.


Solution:

To calculate: base angle

How: Using arctan formula

θ = arctan(opposite ÷ adjacent)

θ = arctan(9 ÷ 17)

= arctan(0.52)

θ = 27.47 degrees or 270

Answer: The angle is 270

 

Example:

Find out the value of θ, given that the base of the triangle ABC is 24 ft and the height is 11 ft


Solution:

arctanθ = opposite / adjacent

arctanθ = 11 ÷ 24 =0.24

arctanθ = 24.60

θ = 240

FAQs (Frequently Asked Questions)

Q1. What is the Use of Arctan?

Answer: The arctan is trigonometric functions and these functions are used to describe values associated with right triangles. Practically, we apply these functions in order to resolve the distance or height of the object which is otherwise complex to measure.


Apart from, it has practical application in different fields like constructing, architecture, landscaping, engineering, physics, and other mathematical and scientific areas.


In addition, this is resolute taking into account the measure of one angle (not right angle) as well the ratio of two sides of the triangle. Apart from, we identify trigonometric functions by the sides of the triangle that are used in the ratio of these formulas:

Sine = opposite / hypotenuse

Cosine = adjacent / hypotenuse

Tangent = opposite / adjacent

Q2. How and Why Do We Use the Inverse of Trigonometric Functions?

Answer: We use the inverse of trigonometric functions in order to resolve the angle measures when we are aware of the sides of the triangle. Moreover, we can also use it to decide the measure of an angle when we are aware of the side opposite and the side adjacent to the angle.


Most considerate, an ideal approach to use the arctan is a scientific calculator. Additionally, we can trace the arctan button that’s just above the tangent on the calculator. Moreover, we can also use a data table to resolve arctan; but this can be an exhaustive and lengthy method, however, it is most feasible if you don’t have a scientific calculator.