Find the exact value of the expression under the given condi

Find the exact value of the expression under the given conditions. It is given that sin alpha = 21/29, 0

Solution

3)

a).

I am taking alpha as \'a\' and beta as \'b\'

so sina = 21/29 a is in QI (first quadrant)

cosb = 3/5 b is also in QI

so we know identity sin^2 x +cos^2 x=1

so sin^2 a +cos^2 a =1

cos^2 a = 1-sin^2a

cos^2 a = 1 - 411/841

cosa =20/29

similer way we can find the value of sinb

sin^2 b = 1-cos^2 b

sin^2b = 1 - 9/25

sinb = 4/5

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

=20/21 . 3/5 - 21/29. 4/5

= 60/105 - 84/105

= -24/105

= -8/35

b).

given tana =20/21 and a is in QIII and cosb= -24/25 b is in QII

from the above problem also tana = 20/21

so we can say sina =-20/29 and cosa= -21/29

sin^2b =1-cos^2b

sin^2b = 1 - 576/625

sinb = 7/25

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

= -20/29.-24/25 + -21/29. 7/25

= 480/725 - 147/725

=333/725