# 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

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.

## Recent Articles

1. ### Types of Fractions |Proper Fraction |Improper Fraction |Mixed Fraction

Mar 02, 24 05:31 PM

The three types of fractions are : Proper fraction, Improper fraction, Mixed fraction, Proper fraction: Fractions whose numerators are less than the denominators are called proper fractions. (Numerato…

2. ### Subtraction of Fractions having the Same Denominator | Like Fractions

Mar 02, 24 04:36 PM

To find the difference between like fractions we subtract the smaller numerator from the greater numerator. In subtraction of fractions having the same denominator, we just need to subtract the numera…

3. ### Addition of Like Fractions | Examples | Worksheet | Answer | Fractions

Mar 02, 24 03:32 PM

To add two or more like fractions we simplify add their numerators. The denominator remains same. Thus, to add the fractions with the same denominator, we simply add their numerators and write the com…

4. ### Comparison of Unlike Fractions | Compare Unlike Fractions | Examples

Mar 01, 24 01:42 PM

In comparison of unlike fractions, we change the unlike fractions to like fractions and then compare. To compare two fractions with different numerators and different denominators, we multiply by a nu…