Question

sin 20 into sin 40 into sin 80 equal to root 3 by 8​

Answer 1

Answer:sin20*sin40*sin80=√3/2

Step-by-step explanation:

Sin20*sin40*sin80

Sin20*sin(60-20)*sin(60+20)

formula:sin(a-b)*sin(a+b)=sin^2(a)-sin^2(b)

sin20[sin^2(60)-sin^2(20)]

sin20[3/4-sin^2(20)]

(sin(20))/4[3-4sin^2(20)]

3sin(20)-4sin^3(20)

formula:3sin(A)-4sin^3(A)=sin3A

sin3(20)

sin60=√3/2

Answer 2

To prove →

sin20° sin 40° sin80° = √3/8 .

Solution→ by method 1st

Taking LHS

(sin20° sin40°) sin80°

Multiple and divide the numerator and denominator by 2 .

$\frac{1}{2} (2 \sin(20) \sin(40) ) \sin(80)$

As we know that 2 sinA sinB = cos ( A -B) - cos ( A + B) So applying this Formula we got:

1/2[cos ( 20° - 40° ) - cos ( 20°+ 40° ) ]sin 80°

1/2 [ cos (-20°) - cos ( 60° )] sin 80°

As we know that cos(-A) = cos A .So ,

1/2 ( cos 20° - cos 60°) sin 80°

1/2 ( cos20°sin 80° - cos60° sin 80°)

cos 60° = 1/2

1/2( cos 20° sin 80° - 1/2 sin 80° )

Again multiple and divide the numerator and denominator by 2:

1/4 ( 2 sin80° cos20° - sin80°)

As we know that 2sinAcosB = sin( A+B)+ sin ( A -B)

1/4 [ sin(80+20) +sin(80-20) - sin 80°]

1/4(sin 100° + sin 60° - sin 80° )

sin60° = √3/2

1/4[sin( 180°-80°)+ √3/4 - sin80°]

1/4 ( sin80° - sin 80° +√3/2)

sin80° cancelled out by - sin80°

1/4(√3/2)

√3/8 = RHS

here we can also use another method.

Solution→ by method 2nd

sin A sin (60-A) sin (60+A ) = sin (3A)/4

So here we can write LHS in above form:

sin20° sin(60°-20°)sin(60°+20°)

Now using the above mentioned formula

sin (20°×3)/4

sin 60°/4

(√3/2)/4

√3/8 = RHS

hence proved.