Find the exact value of each of the following under the give

Find the exact value of each of the following under the given conditions, a. cos (alpha + beta) b. sin (alpha + beta) c. tan (alpha + beta) tanalpha = 3/4, pi

Solution

tana =3/4 a is in QIII

cosb= 2/sqrt(13) b is in QIV

sec^2a -tan^2 a=1

sec^2a = 1+ 9/16

seca = -5/4 [ since a is in QIII]

cosa = -4/5

sina= -3/5

cosb = 2/sqrt(13)

sin^2a +cos^2a =1

sin^2a = 1- 4/13

sina = -3/sqrt(13) (since b is in QIV)

a)cos(a+b) = cosa.cosb - sina.sinb

= -4/5. 2/sqrt(13) - (-3/4). (-3/sqrt(13))

= -8/5sqrt(13) - 9/sqrt(13)

= -17/5sqrt(13)

b).sin(a+b) =sina.cosb +cosa.sinb

= (-3/5). (2/sqrt13) + (-4/5) . (-3/sqrt13)

= -6/5sqrt(13) +12/5sqrt(13)

=6/5sqrt(13)

c).tan(a+b) =sin(a+b)/cos(a+b)

= -6/17