Evaluate cos(a − b)

Question

 if cos a = 4/5 tan a < 0, tan b = −√(15) and cos b < 0.cos (a - b) =?

Answer 1

cos a = 4 /5

tan b =  -

cos (a -b) = cos a cos b + sina sinb

sin a = opposite side /hypotenuse 

cos a = 4 /5 = adjacent side / hypotenuse 

hypotenuse ² = opposite side²   + adjacent side ² 

25 = opposite side² +16 = > opposite side ² = 9 => opposite side = 3.

So, sin a = 3 /5.

Since tana<0  that is tana is negative hence a lies in second quadrant  or fourth quadrant.

tan b = opposite side / adjacent side = -  / 1

hypotenuse ² = opposite side²   + adjacent side ²  = (- )² + 1 = 16

hypotenuse = 4.

sinb = (- ) / 4 .

cos b = 1 /4.

Since cos b < 0 , cosb is negative and hence b lies in second quadrant or third quadrant.

cos(a+b) =  cos a cos b + sin a sinb

                = (4 /5) (1 / 4) + (3 /5)( (- ) / 4) = (1 / 5)  -(3/20) = (4 - 3) /20.

Hence cos (a -b) =  (4 - 3) /20.