Find the exact value of the expression: sin[sin-1 9/41-cos-1(-4/5)]

Question

Please help

Answer 1

sin[sin^-1(9/41)-cos^-1(-4/5)]                  (1)             

 

The first point is that the two terms inside the sine are angles. To solve this I would apply the identity for the difference of two angles. 

 

sin(A-B)=sin(A)cos(B)+cos(A)sin(B)                   

 

For our example we would apply the formula as if: 

A = sin^-1(9/41) and  B=cos^-1(-4/5)

 

The original expression can now be written as: 

 

sin[sin^-1(9/41)]×cos[cos^-1(-4/5)]+cos[ sin^-1(9/41)]×sin[cos^-1(-4/5)]

 

From here you will want to find each factor individually. 

 

 

If we set A = sin^-1(9/41), then: 

 

sin[sin^-1(9/41)] = sin(A)    

 sin(A) =9/41    

 sin[sin^-1(9/41)] = 9/41

 

Similarly: 

B=cos^-1(-4/5)

cos(B)=-4/5      

cos[cos^-1(-4/5)] = -4/5

 

Now we need to find sin(B) and cos(A). Do you see how to do that? If you have sin(A)=9/41 there should be a certain identity that you can use to find the cos(A).