Inverse Trigonometric Functions

%5Csin%20%5E%7B2%7D%20%5Ctheta%20%5Ccos%20%5E%7B3%7D%20%5Ctheta-%5Csin%20%5E%7B4%7D%20%5Ctheta%20%5Ccos%20%5Ctheta

%3D%5Csin%20%5E%7B2%7D%20%5Ctheta%20%5Ccos%20%5Ctheta%5Cleft(%5Ccos%20%5E%7B2%7D%20%5Ctheta-%5Csin%20%5E%7B2%7D%20%5Ctheta%5Cright)

%3D%5Csin%20%5E%7B2%7D%20%5Ctheta%20%5Ccos%20%5Ctheta(%5Ccos%202%20%5Ctheta)

%3D%5Csin%20%5Ctheta%20%5Csin%20%5Ctheta%20%5Ccos%20%5Ctheta(%5Ccos%202%20%5Ctheta)

%3D%5Csin%20%5Ctheta%20%5Ccdot%20%5Cfrac%7B2%7D%7B2%7D%20%5Csin%20%5Ctheta%20%5Ccos%20%5Ctheta(%5Ccos%202%20%5Ctheta)

%3D%5Cfrac%7B1%7D%7B2%7D%20%5Csin%20%5Ctheta%20%5Csin%202%20%5Ctheta%20%5Ccos%202%20%5Ctheta

%5B%5Cbecause%202%20%5Csin%20%5Ctheta%20%5Ccos%20%5Ctheta%3D%5Csin%202%20%5Ctheta%5D

%3D%5Cfrac%7B1%7D%7B2%7D%20%5Csin%20%5Ctheta%20%5Ccdot%20%5Cfrac%7B2%7D%7B2%7D%20%5Csin%202%20%5Ctheta%20%5Ccos%202%20%5Ctheta%3D%5Cfrac%7B1%7D%7B4%7D%20%5Csin%20%5Ctheta%20%5Ccdot%20%5Csin%204%20%5Ctheta

We have,

%5Ccos%5E%7B-1%7D%20%5Cfrac%7B3%7D%7B5%7D%20%2B%20%5Ccos%5E%7B-1%7D%20%5Cfrac%7B12%7D%7B13%7D%20%3D%20%5Ccos%5E%7B-1%7D%5C%2Ck

%5CRightarrow%20%5Ccos%5E%7B-1%7D%5Cleft%5C%7B%5Cfrac%7B3%7D%7B5%7D%20%5Ccdot%20%5Cfrac%7B12%7D%7B13%7D-%5Csqrt%7B1-%5Cleft(%5Cfrac%7B3%7D%7B5%7D%5Cright)%5E%7B2%7D%5Csqrt%7B1-%5Cleft(%5Cfrac%7B12%7D%7B13%7D%5Cright)%5E%7B2%7D%7D%7D%5Cright%5C%7D%20%3D%20%5Ccos%5E%7B-1%7D%5C%2Ck

%5B%5Cbecause%20%5Ccos%5E%7B-1%7D%5C%2Cx%20%2B%20%5Ccos%5E%7B-1%7D%5C%2C%20y%20%3D%20%5Ccos%5E%7B-1%7D%5Cleft%5C%7Bxy-%5Csqrt%7B1-x%5E%7B2%7D%7D%5Csqrt%7B1-y%5E%7B2%7D%7D%5Cright%5C%7D

, if

-1%20%5Cle%20x%2C%20y%20%5Cle%201

and

x%2By%20%5Cge%200%5D

%5CRightarrow%20%5Ccos%5E%7B-1%7D%5Cleft%5C%7B%5Cfrac%7B36%7D%7B65%7D-%5Cfrac%7B4%7D%7B5%7D%20%5Ctimes%20%5Cfrac%7B5%7D%7B13%7D%5Cright%5C%7D%20%3D%20%5Ccos%5E%7B-1%7D%5C%2Ck

%5CRightarrow%20%5Ccos%5E%7B-1%7D%5Cleft%5C%7B%5Cfrac%7B36%7D%7B65%7D-%5Cfrac%7B20%7D%7B65%7D%5Cright%5C%7D%20%3D%20%5Ccos%5E%7B-1%7D%5C%2Ck

%5CRightarrow%20%5Ccos%5E%7B-1%7D%20%5Cfrac%7B16%7D%7B65%7D%20%3D%20%5Ccos%5E%7B-1%7D%5C%2Ck

%5CRightarrow%20k%3D%5Cfrac%7B16%7D%7B65%7D

Chapter Questions
2 MCQs