General and Principal Values of tan\(^{-1}\) x

How to find the general and principal values of tan\(^{-1}\) x?

Let tan θ = x (- ∞ < x < ∞) then θ = tan\(^{-1}\) x.

Here θ has infinitely many values.

Let – \(\frac{π}{2}\) < α < \(\frac{π}{2}\), where α is positive or negative smallest numerical value of these infinite number of values and satisfies the equation tan θ = x then the angle α is called the principal value of tan\(^{-1}\) x.

Again, if the principal value of tan\(^{-1}\) x is α (– \(\frac{π}{2}\) < α < \(\frac{π}{2}\)) then its general value = nπ + α.

Therefore, tan\(^{-1}\) x = nπ + α, where, (– \(\frac{π}{2}\) < α < \(\frac{π}{2}\)) and (- ∞ < x < ∞).


Examples to find the general and principal values of arc tan x:

1. Find the General and Principal Values of tan\(^{-1}\) (√3).

Solution:

Let x = tan\(^{-1}\) (√3)

⇒ tan x = √3

⇒ tan x = tan \(\frac{π}{3}\)

⇒ x = \(\frac{π}{3}\)

⇒ tan\(^{-1}\) (√3) = \(\frac{π}{3}\)

Therefore, principal value of tan\(^{-1}\) (√3) is \(\frac{π}{3}\) and its general value = nπ + \(\frac{π}{3}\).

 

2. Find the General and Principal Values of tan\(^{-1}\) (- √3)

Solution:

Let x = tan\(^{-1}\) (-√3)

⇒ tan x = -√3

⇒ tan x = tan (-\(\frac{π}{3}\))

⇒ x = -\(\frac{π}{3}\)

⇒ cos\(^{-1}\) (-√3) = -\(\frac{π}{3}\)

Therefore, principal value of tan\(^{-1}\) (-√3) is -\(\frac{π}{3}\) and its general value = nπ - \(\frac{π}{3}\).





11 and 12 Grade Math

From General and Principal Values of arc tan x  to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.