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:

sin = 4/5, 0<a</2 ; cos = (4*sqrt(65)) / 65, -/2<<0

(A) sin ( + )  

(b) cos ( + )

(c) sin( - )

(d) tan( - )

Solution

Let alpha = a, beta= b

Sin a= 4/5 Cos a = (1- sin² a)^1/2=(9/25)^1/2=3/5

Cos b= (4× (65)^1/2)/65 =( 4×8.06)/65= 32.24/65= 0.4961

Cos b~0.5

Cos b = 0.5= 5/10

Sin b = (1-cos²b)^1/2= (75/100)^1/2= (5 (3)^1/2)/10

Sin (a+b)= sin a× cos b +cos a × sin b

=(4/5)×(5/10)+(3/5)×(5/10)×(3)^1/2 = (4+3^3/2)/10

Cos (a+b)= cos a × cos b- sin a × sin b

= (3/5)×(5/10) - (4/5)×(5/10)(3)^1/2

=(3-4 (3)^1/2)/10

Sin (a-b)= sin a× cos b - cos a× sin b

=(4/5)×(5/10) -(3/5)×(5/10)(3)^1/2

= (4-3^3/2)/10

tan (a-b)= sin (a-b)/ cos (a-b)

cos (a-b)= cos a×cos b+Sin a ×sin b

=(3/5)×(5/10)+(4/5)×(5/10)(3)^1/2

=(3+4(3)^1/2)/10

tan (a-b)= sin (a-b)/cos (a-b)

=((4-3^3/2)/10)/((3+4 (3)^1/2)/10)

=(4-3^3/2)/(3+4 (3)^1/2)