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 - \(\frac{π}{3}\).

● Inverse Trigonometric Functions

• General and Principal Values of sin\(^{-1}\) x
• General and Principal Values of cos\(^{-1}\) x
• General and Principal Values of tan\(^{-1}\) x
• General and Principal Values of csc\(^{-1}\) x
• General and Principal Values of sec\(^{-1}\) x
• General and Principal Values of cot\(^{-1}\) x
• Principal Values of Inverse Trigonometric Functions
• General Values of Inverse Trigonometric Functions
  
• arcsin(x) + arccos(x) = \(\frac{π}{2}\)
• arctan(x) + arccot(x) = \(\frac{π}{2}\)
• arctan(x) + arctan(y) = arctan(\(\frac{x + y}{1 - xy}\))
• arctan(x) - arctan(y) = arctan(\(\frac{x - y}{1 + xy}\))
• arctan(x) + arctan(y) + arctan(z)= arctan\(\frac{x + y + z – xyz}{1 – xy – yz – zx}\)
• arccot(x) + arccot(y) = arccot(\(\frac{xy - 1}{y + x}\))
• arccot(x) - arccot(y) = arccot(\(\frac{xy + 1}{y - x}\))
• arcsin(x) + arcsin(y) = arcsin(x \(\sqrt{1 - y^{2}}\) + y\(\sqrt{1 - x^{2}}\))
• arcsin (x) - arcsin(y) = arcsin (x \(\sqrt{1 - y^{2}}\) - y\(\sqrt{1 - x^{2}}\))
• arccos (x) + arccos(y) = arccos(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))
• arccos(x) - arccos(y) = arccos(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))
• 2 arcsin(x) = arcsin(2x\(\sqrt{1 - x^{2}}\)) 
• 2 arccos(x) = arccos(2x\(^{2}\) - 1)
• 2 arctan(x) = arctan(\(\frac{2x}{1 - x^{2}}\)) = arcsin(\(\frac{2x}{1 + x^{2}}\)) = arccos(\(\frac{1 - x^{2}}{1 + x^{2}}\))
• 3 arcsin(x) = arcsin(3x - 4x\(^{3}\))
• 3 arccos(x) = arccos(4x\(^{3}\) - 3x)
• 3 arctan(x) = arctan(\(\frac{3x - x^{3}}{1 - 3 x^{2}}\))
• Inverse Trigonometric Function Formula
• Principal Values of Inverse Trigonometric Functions
• Problems on Inverse Trigonometric Function

11 and 12 Grade Math

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