Inverse Trigonometric Functions

Let sin⁻¹ $$ $\left(\frac{-1}{2}\right)$ =y. Then, sin y=- $\frac{-1}{2}$

We know that the range of principal value branch of sin⁻¹ is $-\frac{\pi }{2},\;$

and sin⁻¹ $\left(\frac{-1}{2}\right)$ =−sin⁻¹$\frac{-1}{2}$ (sin (-x) = -sin x)

= $\left(-\frac{\pi }{6}\right)$ = sin y (as sin
$\frac{\pi }{6}$
$\frac{}{}$ = $\frac{1}{2}$ )

Principal value of sin⁻¹ $\left(-\frac{1}{2}\right)$ is $\left(-\frac{\pi }{6}\right)$

students can check the table for the principal values for all ITFs below;

To understand this, Assume you have a bucket that has infinite number of apples and if your mother asks "give me the apple". How will you figure out which one is "The Apple", she is asking for.

Similarly any inversre trigonometric functions behaves like a Many-one Function; which means,

For Example $\sin^ {-1}\left (\frac {2} {3}\right)$ can have many solutions, we need to fix one solutions which can be used as standerd value for the function.

• A standerd value (Angles) of any inverse trigonometric value lies between a fiexed range is known as principal value. 

For Ex; The value of $\sin^ {-1}\left (\frac {2} {3}\right)$ will always lie between –? /2 to? /2.

$y={\sin }^{?1}\left(\frac{2}{3}\right)?\sin \left(y\right)=\frac{2}{3}?y?[?\frac{?}{2},\frac{?}{2}]$  

Inverse Trigonometric Functions is considered a moderated weightage topic, 1 question is expected from this chapter directly. However, ITF plays an important role to boost score through usgae of its properties in Calculus unit, which have a better weightage overall. Students must check the NCERT Solutions of Inverse Trigonometric functions to build a strong base. Students can check the NCERT ITF Solutions below;

Class 12 Math ITF Solutions

There are six basic trigonometric functions like sin, cos, tan, cot, sec, and cosec. These basic trigonometric functions are used to find the ratio of 2 sides among the three sides.  Inverse trigonometric functions as suggested by names are inverse of trigonometric functions, are used to find angles when the trigonometric ratio is given. For more information about the Class 12 ITF chapter, check the link below;

Class 12 Inverse Trigonometric Functions

Class 12 Inverse Trigonometric Functions is a topic of moderate weightage but its application are far more important in calculus which have very high weightage around 35-40 marks. Students can expect up to 4 marks from ITF chapter. Students can expect 1 MCQ/Very Short Answer type question of 1 mark, 1-2 short answer type question of 2 mark and ocassionaly long answer type question of 4 marks from this chapter. For more details candidates can check below link;

Class 12 ITF NCERT Solutions