Trigonometric Functions

We have,

%5Cfrac%7B7%5Cpi%7D%7B8%7D%3D%5Cpi-%5Cfrac%7B%5Cpi%7D%7B8%7D

and

%5Cfrac%7B5%5Cpi%7D%7B8%7D%3D%5Cpi-%5Cfrac%7B3%5Cpi%7D%7B8%7D

%5CRightarrow%20%5Csin%20%5Cfrac%7B7%5Cpi%7D%7B8%7D%3D%5Csin%5Cleft(%5Cpi-%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)

and

%5Csin%20%5Cfrac%7B5%5Cpi%7D%7B8%7D%3D%5Csin%5Cleft(%5Cpi-%5Cfrac%7B3%5Cpi%7D%7B8%7D%5Cright)

%5CRightarrow%20%5Csin%20%5Cfrac%7B7%5Cpi%7D%7B8%7D%3D%5Csin%20%5Cfrac%7B%5Cpi%7D%7B8%7D

and

%5Csin%20%5Cfrac%7B5%5Cpi%7D%7B8%7D%3D%5Csin%20%5Cfrac%7B3%5Cpi%7D%7B8%7D

%5CRightarrow%20%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B7%5Cpi%7D%7B8%7D%5Cright)%3D%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)

and

%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B5%5Cpi%7D%7B8%7D%5Cright)%3D%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B8%7D%5Cright)

Now,

%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)%2B%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B8%7D%5Cright)%2B%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B5%5Cpi%7D%7B8%7D%5Cright)%2B%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B7%5Cpi%7D%7B8%7D%5Cright)

%3D%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)%2B%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B8%7D%5Cright)%2B%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B8%7D%5Cright)%2B%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)

%3D2%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)%2B2%5Csin%5E%7B4%7D%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B8%7D%5Cright)

%3D2%5Cleft%5B%5Cleft(%5Csin%5E%7B2%7D%20%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)%5E%7B2%7D%2B%5Cleft(%5Csin%5E%7B2%7D%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B8%7D%5Cright)%5Cright)%5E%7B2%7D%5Cright%5D

%3D2%5Cleft%5B%5Cleft(%5Cfrac%7B1-%5Ccos%5C%2C2%5Cleft(%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)%7D%7B2%7D%5Cright)%5E%7B2%7D%2B%5Cleft(%5Cfrac%7B1-%5Ccos%5C%2C2%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B8%7D%5Cright)%7D%7B2%7D%5Cright)%5E%7B2%7D%5Cright%5D

%3D2%5Cleft%5B%5Cfrac%7B%5Cleft(1-%5Ccos%20%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)%5E%7B2%7D%7D%7B4%7D%2B%5Cfrac%7B%5Cleft(1-%5Ccos%20%5Cfrac%7B3%5Cpi%7D%7B4%7D%5Cright)%5E%7B2%7D%7D%7B4%7D%5Cright%5D

%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cleft(1-%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cright)%5E%7B2%7D%2B%5Cleft(1%2B%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cright)%5E%7B2%7D%5Cright%5D

%5Cleft%5B%5Cbecause%20%5Ccos%20%5Cfrac%7B%5Cpi%7D%7B4%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5C%2C%5Ctext%7Band%7D%20%5C%2C%5Ccos%20%5Cfrac%7B3%5Cpi%7D%7B4%7D%3D%5Cfrac%7B-1%7D%7B%5Csqrt%7B2%7D%7D%5Cright%5D

%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B1%2B%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B2%7D%7B%5Csqrt%7B2%7D%7D%2B1%2B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B2%7D%7B%5Csqrt%7B2%7D%7D%5Cright%5D

%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B2%2B%5Cfrac%7B2%7D%7B2%7D%5Cright%5D%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B2%2B1%5Cright%5D%3D%5Cfrac%7B3%7D%7B2%7D

Chapter Questions
3 MCQs