Question

18. In a triangle ABC, if the median AD makes an angle theta with BC and AB = 2 AD then sin theta=
a) 2sin A
b) 2 sin B
c) 2sin C
d) sin A + sin B​

Answer 1

Answer:

Let ∠ADC=α

Using Sine law in ΔADC we get;

sin(A−θ)2a=sin αb       (1)

Using Sine law in ΔABD we get;

c/sinθ2a=sin(π−α)⟹sinθ2a=sinαc

⟹sinα=2acsinθ

Putting the value of sinα in equation 1; we get

sin(A−θ)(2a)=csinθb(2a)

⟹sin(A−θ)=bcsinθ

Hence; sin(A−θ)=bcsinθ

Answer 2

Let ∠ADC=α

Using Sine law in ΔADC we get;

sin(A−θ)

2

a

=

sinα

b

(1)

Using Sine law in ΔABD we get;

sinθ

2

a

=

sin(π−α)

c

⟹

sinθ

2

a

=

sinα

c

⟹sinα=

2

a

c

sinθ

Putting the value of sinα in equation 1; we get

sin(A−θ)

(

2

a

)

=

csinθ

b(

2

a

)

⟹sin(A−θ)=

b

c

sinθ

Hence; sin(A−θ)=

b

c

sinθ