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General and Principal Values of csc1  x

How to find the general and principal values of ccs1 x?

Let csc θ = x (| x |≥ 1 i.e., x ≥ 1 or, x ≤ - 1) then θ = csc1 x .

Here θ has infinitely many values.

Let – π2 ≤ α ≤ π2, where α is non-zero(α ≠ 0) positive or negative smallest numerical value of these infinite number of values and satisfies the equation csc θ = x then the angle α is called the principal value of csc1 x.

Again, if the principal value of csc1 x is α (– π2 < α < π2) and α ≠ 0 then its general value = nπ + (- 1) n α, where, | x | ≥ 1.

Therefore, tan1 x = nπ + α, where, (– π2 < α < π2), | x | ≥ 1 and (- ∞ < x < ∞).

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

1. Find the General and Principal Values of csc 1 (√2).

Solution:

Let x = csc1 (√2)

⇒ csc x = √2

⇒ csc x = csc π4

⇒ x = π4

⇒ csc1 (√2) = π4

Therefore, principal value of csc1 (√2) is π4 and its general value = nπ + (- 1)nπ4.

 

2. Find the General and Principal Values of csc 1 (-√2).

Solution:

Let x = csc1 (-√2)

⇒ csc x = -√2

⇒ csc x = csc (-π4)

⇒ x = -π4

⇒ csc1 (-√2) = -π4

Therefore, principal value of csc1 (-√2) is -π4 and its general value = nπ + (- 1)n ∙ (-π4) = nπ - (- 1)nπ4.

 Inverse Trigonometric Functions







11 and 12 Grade Math

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