Principal Values of Inverse Trigonometric Functions

We will learn how to find the principal values of inverse trigonometric functions in different types of problems.

The principal value of sin\(^{-1}\) x for x > 0, is the length of the arc of a unit circle centred at the origin which subtends an angle at the centre whose sine is x. For this reason sin^-1 x is also denoted by arc sin x. Similarly, cos\(^{-1}\) x, tan\(^{-1}\)  x, csc\(^{-1}\)  x, sec\(^{-1}\)  x and cot\(^{-1}\) x are denoted by arc cos x, arc tan x, arc csc x, arc sec x.

1. Find the principal values of sin\(^{-1}\) (- 1/2)      

Solution: 

If θ be the principal value of sin\(^{-1}\) x then - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\).

Therefore, If the principal value of sin\(^{-1}\) (- 1/2) be θ then sin\(^{-1}\) (- 1/2) = θ

⇒ sin θ = - 1/2 = sin (-\(\frac{π}{6}\)) [Since, - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\)]

Therefore, the principal value of sin\(^{-1}\) (- 1/2) is (-\(\frac{π}{6}\)).



2. Find the principal values of the inverse circular function cos\(^{-1}\) (- √3/2)

Solution:

 If the principal value of cos\(^{-1}\) x is θ then we know, 0 ≤ θ ≤ π.

Therefore, If the principal value of cos\(^{-1}\)  (- √3/2) be θ then cos\(^{-1}\)  (- √3/2) = θ

⇒ cos θ = (- √3/2) = cos \(\frac{π}{6}\) = cos (π - \(\frac{π}{6}\)) [Since, 0 ≤ θ ≤ π]

Therefore, the principal value of cos\(^{-1}\)  (- √3/2) is π - \(\frac{π}{6}\) = \(\frac{5π}{6}\).

 

3. Find the principal values of the inverse trig function tan\(^{-1}\) (1/√3)

Solution:

If the principal value of tan\(^{-1}\) x is θ then we know, - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\).

Therefore, If the principal value of tan\(^{-1}\) (1/√3) be θ then tan\(^{-1}\) (1/√3) = θ

⇒ tan θ = 1/√3 = tan \(\frac{π}{6}\) [Since, - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\)]

Therefore, the principal value of tan\(^{-1}\) (1/√3) is \(\frac{π}{6}\).

 

4. Find the principal values of the inverse circular function cot\(^{-1}\) (- 1)

Solution:

If the principal value of cot\(^{-1}\) x is α then we know, - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\) and θ ≠ 0.

Therefore, If the principal value of cot\(^{-1}\) (- 1) be α then cot\(^{-1}\) (- 1) = θ

⇒ cot θ = (- 1) = cot (-\(\frac{π}{4}\)) [Since, - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\)]  

Therefore, the principal value of cot\(^{-1}\) (- 1) is (-\(\frac{π}{4}\)).    

 

5. Find the principal values of the inverse trig function sec\(^{-1}\) (1)

Solution:

If the principal value of sec\(^{-1}\) x is α then we know, 0 ≤ θ ≤ π and θ ≠ \(\frac{π}{2}\).

Therefore, If the principal value of sec\(^{-1}\) (1) be α then, sec\(^{-1}\) (1) = θ

⇒ sec θ = 1 = sec 0    [Since, 0 ≤ θ ≤ π]

Therefore, the principal value of sec\(^{-1}\) (1) is 0.

 

6. Find the principal values of the inverse trig function csc\(^{-1}\) (- 1).

Solution:

If the principal value of csc\(^{-1}\) x is α then we know, - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\) and θ ≠ 0.

Therefore, if the principal value of csc\(^{-1}\) (- 1) be θ then csc\(^{-1}\) (- 1) = θ

⇒ csc θ = - 1 = csc (-\(\frac{π}{2}\)) [Since, - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\)]

Therefore, the principal value of csc\(^{-1}\) (- 1) is (-\(\frac{π}{2}\)).

 Inverse Trigonometric Functions




11 and 12 Grade Math

From Principal Values of Inverse Trigonometric Functions 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. Word Problems on Dividing Money | Solving Money Division Word Problems

    Feb 13, 25 10:29 AM

    Word Problems on Dividing Money
    Read the questions given in the word problems on dividing money. We need to understand the statement and divide the amount of money as ordinary numbers with two digit numbers. 1. Ron buys 15 pens for…

    Read More

  2. Addition and Subtraction of Money | Examples | Worksheet With Answers

    Feb 13, 25 09:02 AM

    Add Money Method
    In Addition and Subtraction of Money we will learn how to add money and how to subtract money.

    Read More

  3. Worksheet on Division of Money | Word Problems on Division of Money

    Feb 13, 25 03:53 AM

    Division of Money Worksheet
    Practice the questions given in the worksheet on division of money. This sheet provides different types of questions on dividing the amount of money by a number; finding the quotient

    Read More

  4. Worksheet on Multiplication of Money | Word Problems | Answers

    Feb 13, 25 03:17 AM

    Worksheet on Multiplication of Money
    Practice the questions given in the worksheet on multiplication of money. This sheet provides different types of questions on multiplying the amount of money by a number; arrange in columns the amount…

    Read More

  5. Division of Money | Worked-out Examples | Divide the Amounts of Money

    Feb 13, 25 12:16 AM

    Divide Money
    In division of money we will learn how to divide the amounts of money by a number. We carryout division with money the same way as in decimal numbers. We put decimal point in the quotient after two pl…

    Read More