Use the formula 

    cos(a-b) = cos(a)*cos(b) + sin(a)*sin(b)    (1)


Regarding this formula, see the lesson Addition and subtraction formulas in this site.


In addition to the given  sin(a) =   and  cos(b) = , you need to know  cos(a) and sin(b).



    1.  cos(a) =  =  =  =  =  = .

        The sign "+" was chosen at the square root because the angle "a" is in QI.


    2.  sin(b) =  =  =  =  =  = .

        The sign "+" was chosen at the square root because sin(b) is positive when the angle "b" is in QII.



Now all you need to do is to substitute everything into the formula (1) and make the calculations.


cos(a-b) =  =  =  = .     ANSWER

The formula is .

Draw angle α in Quadrant I:

Since sine = , we make y=4 and r=5, so that the sin(α)
will be .



For the formula, we need sine and cosine, and the cosine is 

So we find x by the Pythagorean relation: 







Since x goes to the right, we know to take the positive
value .  So now we know that 

Next we draw angle β in Quadrant II:

Since cosine = , we make x=-5 and r=13, so that the cos(β)
will be .



For the formula, we need sine and cosine, and the sine is 

So we find x by the Pythagorean relation: 







Since y goes up from the x-axis, we know to take the positive
value .  So now we know that 

Now we use the formula

.







Edwin