Question

Question 10: Find the principal value of cosec¯¹(√2)

Class 12 - Math - Inverse Trigonometric Functions

Answer 1

Let −¹ (−√2) = ∅,
then ∅= −√2 =− ( /4 )
= (− /4 )

We know that the range of the principal value branch of cosec−¹ is [- π /2 , π /2 ]-{0}
and cosec (- π /4 ) = -√2.
Therefore,
the principal value of cosec-¹(-√2) is - π /4

Answer 2

The principal value of $cosec^{-1} \sqrt{2}$ is  $-\frac{\pi }{4}$

Step-by-step explanation:

Given : $cosec^{-1} \sqrt{2}$

To find : The principal value of $cosec^{-1} \sqrt{2}$

Solution:

• Inverse trigonometric functions are closely related to basic trigonometric ratios
• The range and domain of the inverse trigonometric function are replaced with the domain (value) and range (response) of the trigonometric ratio.

Let we take,

$cosec^{-1} \sqrt{2} = y$

$cosec y = \sqrt{2}$

⇒ - cosec $(\frac{\pi }{4} )$

⇒ cosec $( - \frac{\pi }{4})$

Range of the principal value of $cosec^{-1} x$

= $[\frac{-\pi }{2} ,\frac{-\pi }{2} ] - [0]$

Therefore the principal value is ($-\frac{\pi }{4}$)

Final answer:

The principal value of $cosec^{-1} \sqrt{2}$ is  $-\frac{\pi }{4}$

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