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

From General and Principal Values of arc sec 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.




Share this page: What’s this?

Recent Articles

  1. Multiplication by Ten, Hundred and Thousand |Multiply by 10, 100 &1000

    May 01, 25 11:57 PM

    Multiply by 10
    To multiply a number by 10, 100, or 1000 we need to count the number of zeroes in the multiplier and write the same number of zeroes to the right of the multiplicand. Rules for the multiplication by 1…

    Read More

  2. Adding and Subtracting Large Decimals | Examples | Worksheet | Answers

    May 01, 25 03:01 PM

    Here we will learn adding and subtracting large decimals. We have already learnt how to add and subtract smaller decimals. Now we will consider some examples involving larger decimals.

    Read More

  3. Converting Fractions to Decimals | Solved Examples | Free Worksheet

    Apr 28, 25 01:43 AM

    Converting Fractions to Decimals
    In converting fractions to decimals, we know that decimals are fractions with denominators 10, 100, 1000 etc. In order to convert other fractions into decimals, we follow the following steps:

    Read More

  4. Expanded Form of a Number | Writing Numbers in Expanded Form | Values

    Apr 27, 25 10:13 AM

    Expanded Form of a Number
    We know that the number written as sum of the place-values of its digits is called the expanded form of a number. In expanded form of a number, the number is shown according to the place values of its…

    Read More

  5. Converting Decimals to Fractions | Solved Examples | Free Worksheet

    Apr 26, 25 04:56 PM

    Converting Decimals to Fractions
    In converting decimals to fractions, we know that a decimal can always be converted into a fraction by using the following steps: Step I: Obtain the decimal. Step II: Remove the decimal points from th…

    Read More