Find the exact value under the given conditions sin 35 0

Find the exact value under the given conditions:

sin= 3/5 ,0<<pi/2

cos=2sqrt(5)/5, -pi/2<<pi/2

a) sin(+)

b) cos(+)

c) sin ()

d) tan()

Solution

I am using A for alpa and B for beta as I can not type Alpha and Beta

SinA = 3/5

opp/hyp = 3/5; opp²/hyp² = 9/25

Using Pythogiris theoram opp² + adj² = hyp²

9 + adj² = 25

adj² = 25 -9 = 16

adj = sqrt(16) = 4

cosA = adj/hyp = 4/5

TanA = SinA/CosA = (3/5)/(4/5) = 3/4

CosB = 2sqrt(5)/5 = 2/sqrt(5)

adj/hyp = 2/sqrt(5); adj²/hyp² = 4/5

opp² + adj² = hyp²

opp² + 4 = 5

opp² = 1

opp = 1

SinB = 1/Sqrt(5)

TanB = (1/sqrt(5))/(4/sqrt(5) = 1/4

SinA = 3/5; CosA=4/5; TanA = 3/4

SinB = 1/sqrt(5); CosB = 2/sqrt(5); TanB = 1/4

a) using Double angle formula

Sin(A+B) = SinACosB + CosASinB

= (3/5)(2/sqrt(5) + (4/5)(1/sqrt(5)

= 6/5sqrt(5) + 4/5sqrt(5)

= 10/5sqrt(5) = 2/sqrt(5)

b) Using double angle formula

Cos(A+B) = CosACosB - SinASinB

= (4/5)(2/sqrt(5) - (3/5)(1/sqrt(5)

= 8/5sqrt(5) - 3/5sqrt(5)

= 5/5sqrt(5) = 1/5sqrt(5)

C) using double angle formula

Sin(A-B) = SinACosB - CosASinB

= (3/5)(2/sqrt(5) - (4/5)(1/sqrt(5)

= 6/5sqrt(5) - 4/5sqrt(5)

= 2/5sqrt(5)

d) Using double angle formula

Tan(A-B) = (TanA - TanB)/(1+TanATanB)

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

= (2/4)/(1+3/16)

= (1/2)/(19/16)

= (1/2)(16/19) = 8/19