If `cos A = -5/13`, what is `cos (A - pi/4)` ? It wants the answer in exact form.

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There is a formula `cos(a-b)=cos(a)cos(b)+sin(a)sin(b).`

In our case, `a=A` and `b=pi/4,` and we know `cos(pi/4)=sin(pi/4)=sqrt(2)/2.` So 

`cos(A-pi/4) = cos(A)cos(pi/4)+sin(A)sin(pi/4)=sqrt(2)/2(cos(A)+sin(A)).`

 

`cos(A)` is given, what about `sin(A)?` Of course `cos^2(A)+sin^2(A)=1,` so

`sin(A) = +-sqrt(1-cos^2(A))=+-sqrt(1-25/169) = +-sqrt(144/169) = +-12/13.`

 

To select "+" or "-" we have to know something additional about `A.` If it resides in the II quadrant, then "+", if in the III quadrant, then "-" (it cannot reside in the I or IV quadrants because its cosine is negative).

Without any additional information, we can only state that the answer is either  `sqrt(2)/2*7/13`  or  `-sqrt(2)/2*7/13.`

 

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