In last page,we cover about what are Inverse Trigonometric Functions and how they are used to find the angles corresponding to given trigonometric ratios, which is an essential concept in mathematics, physics, and engineering
In this article, we will discuss the properties of these inverse trigonometric functions.
It may be mentioned here that these results are valid within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined.
Some results may not be valid for all values of the domains of inverse trigonometric functions
Inverse of Negative x
sin−1(−x)=−sin−1(x) cos−1(−x)=π−cos−1(x) tan−1(−x)=−tan−1(x) sec−1(−x)=π−sec−1(x) cosec−1(−x)=−cosec−1(x) cot−1(−x)=π−cot−1(x)
Lets see the proof of one of them Proof on Sin
Let sin−1(−x)=y
Then −x=siny
or x=−siny
or x=sin(−y)
Hence sin−1x=−y=−sin−1(−x)
Similary we can prove for others
sin−1(x)+sin−1(y)=sin−1(x√1−y2+y√1−x2) if x,y≥0, x2+y2≤1 sin−1(x)+sin−1(y)=π−sin−1(x√1−y2+y√1−x2) if x,y≥0, x2+y2>1 sin−1(x)−sin−1(y)=sin−1(x√1−y2−y√1−x2) if x,y≥0, x2+y2≤1 sin−1(x)−sin−1(y)=π−sin−1(x√1−y2−y√1−x2) if x,y≥0, x2+y2>1 cos−1(x)+cos−1(y)=cos−1(xy−√1−y2√1−x2) if x,y≥0, x2+y2≤1 cos−1(x)+cos−1(y)=π−cos−1((xy−√1−y2√1−x2) if x,y≥0, x2+y2>1 cos−1(x)−cos−1(y)=cos−1(xy+√1−y2√1−x2) if x,y≥0, x2+y2≤1 cos−1(x)−cos−1(y)=π−cos−1(xy+√1−y2√1−x2) if x,y≥0, x2+y2>1 tan−1(x)+tan−1(y)=tan−1(x+y1−xy) , if x,y>0, xy<1 tan−1(x)+tan−1(y)=π+tan−1(x+y1−xy) , if x,y>0, xy>1 tan−1(x)+tan−1(y)=tan−1(x+y1−xy)−π , if x<0,y>0, xy>1 tan−1(x)−tan−1(y)=tan−1(x−y1+xy)−π , if xy>−1 tan−1(x)+tan−1(y)+tan−1(z)=tan−1(x+y+z−xyz1−xy−yz−xz)
Double angle identity for Inverse functions
2sin−1(x)=sin−1(2x√1−x2) if −1√2≤x1√2 2cos−1(x)=cos−1(2x2−1) 2tan−1(x)=tan−1(2x1−x2) if $ -1 2tan−1(x)=sin−1(2x1+x2) if |x|≤1 2tan−1(x)=cos−1(1−x21+x2) if x≥0 3sin−1(x)=sin−1(3x−4x3) 3cos−1(x)=cos−1(4x3−3x) 3tan−1(x)=tan−1(3x−x31−3x2)
Odd and Even Properties
Arcsine and arctangent are odd functions sin−1(−x)=−sin−1(x) tan−1(−x)=−tan−1x
