Prove that:
sin^2(20).sin^2(40).sin^2(80)=3/2
Hard to prove something which is not true
LS= (.11697..)(.41317..)(.9698..)
= appr .0469
≠ 3/2
(I assumed your . meant multiplication, and you were in degrees.
It is also false in radians)
It is in degrees m sorry but stil i dnt knw how to prove this
But I showed you that it is NOT true, so you can't prove it to be true.
To prove the equation sin^2(20) * sin^2(40) * sin^2(80) = 3/2, we can use trigonometric identities and properties to simplify the equation and then evaluate both sides to see if they are equal.
Step 1: Convert angles to degrees
The given equation involves three angles: 20 degrees, 40 degrees, and 80 degrees. We will work with degrees for simplicity.
Step 2: Use double-angle formula
The double-angle formula for sine states that sin(2θ) = 2sin(θ)cos(θ). We can use this formula to simplify the equation by expressing each of the angles in terms of half-angle or double-angle relationships.
sin^2(20) = sin^2(2*10) = (2sin(10)cos(10))^2 = 4sin^2(10)cos^2(10)
sin^2(40) = sin^2(2*20) = (2sin(20)cos(20))^2 = 4sin^2(20)cos^2(20)
sin^2(80) = sin^2(2*40) = (2sin(40)cos(40))^2 = 4sin^2(40)cos^2(40)
Step 3: Substitute simplified expressions
Now we will substitute the simplified expressions into the original equation:
4sin^2(10)cos^2(10) * 4sin^2(20)cos^2(20) * 4sin^2(40)cos^2(40) = 3/2
Simplifying:
64sin^2(10)sin^2(20)sin^2(40)cos^2(10)cos^2(20)cos^2(40) = 3/2
Step 4: Apply the sine double-angle formula
Now we can use the double-angle formula again, but this time for cosine, to get rid of the squared terms.
cos(2θ) = cos^2(θ) - sin^2(θ)
cos^2(10) = (1 - sin^2(10))
cos^2(20) = (1 - sin^2(20))
cos^2(40) = (1 - sin^2(40))
Substituting the above expressions into our equation:
64sin^2(10)sin^2(20)sin^2(40)(1 - sin^2(10))(1 - sin^2(20))(1 - sin^2(40)) = 3/2
Step 5: Simplifying the equation further
Expand and simplify the equation:
64sin^2(10)sin^2(20)sin^2(40)(1 - sin^2(10) - sin^2(20) + sin^2(10)sin^2(20))(1 - sin^2(40)) = 3/2
64sin^2(10)sin^2(20)sin^2(40)(1 - sin^2(10) - sin^2(20) + sin^2(10)sin^2(20) - sin^2(40) + sin^2(10)sin^2(40) + sin^2(20)sin^2(40) - sin^2(10)sin^2(20)sin^2(40)) = 3/2
Notice that many terms will cancel out when we expand the equation. By simplifying further, we obtain:
64sin^2(10)sin^2(20)sin^2(40) - 64sin^2(10)sin^2(20)sin^2(40)^2 = 3/2
Finally, we notice that sin^2(40) = 1 - cos^2(40) (using the Pythagorean identity), so we substitute that into the equation:
64sin^2(10)sin^2(20)(1 - cos^2(40)) - 64sin^2(10)sin^2(20)(1 - cos^2(40))^2 = 3/2
Now we have an equation that only involves sine and cosine functions. We can solve this equation by calculating both sides separately to see if they are equivalent.