How do we show (sin 10)^2 + (sin 50)^2 + (sin 70)^2 = 3/2 ???
I can't even think of a way to start this..
since sin^2θ = (1-cos2θ)/2, we have
(1-cos20)/2 + (1-cos100)/2 + (1-cos140)/2 = 3/2
cos20+cos100+cos140 = 0
Using the sum-to-product rules,
cos20 + cos100+cos140
= cos20 + 2cos120cos20
= cos20 - cos20
= 0
Thank you very much Steve
To prove the equation (sin 10)^2 + (sin 50)^2 + (sin 70)^2 = 3/2, we will utilize the trigonometric identity: sin^2θ + cos^2θ = 1.
1. Start by expanding the equation using this identity:
(sin 10)^2 + (sin 50)^2 + (sin 70)^2 = (sin 10)^2 + (1 - cos^2 50) + (1 - cos^2 70)
2. Simplify the equation further:
(sin 10)^2 + (sin 50)^2 + (sin 70)^2 = sin^2 10 + (1 - cos^2 50) + (1 - cos^2 70)
= sin^2 10 + 1 - cos^2 50 + 1 - cos^2 70
3. Now, let's deal with each term separately.
a) For sin^2 10:
We know that sin^2θ = 1 - cos^2θ, so sin^2 10 = 1 - cos^2 10.
b) For cos^2 50:
Since cos^2θ + sin^2θ = 1, we can rewrite cos^2 50 = 1 - sin^2 50.
c) For cos^2 70:
Similarly, cos^2 70 = 1 - sin^2 70.
Substitute these values back into the equation.
4. Substitute the simplified values back into the equation:
(sin 10)^2 + (sin 50)^2 + (sin 70)^2 = (1 - cos^2 10) + 1 - (1 - sin^2 50) + 1 - (1 - sin^2 70)
= 1 - cos^2 10 + 1 - 1 + sin^2 50 + 1 - 1 + sin^2 70
5. Combine like terms:
(sin 10)^2 + (sin 50)^2 + (sin 70)^2 = 3 - cos^2 10 + sin^2 50 + sin^2 70
6. Now, we need to establish a connection between the angles 10, 50, and 70. Notice that sin 10 is equal to cos 80 and sin 70 is equal to cos 20.
Substitute these values back into the equation:
(sin 10)^2 + (sin 50)^2 + (sin 70)^2 = 3 - cos^2 10 + sin^2 50 + sin^2 70
= 3 - cos^2 10 + cos^2 80 + cos^2 20
7. Apply the identity cos^2θ + sin^2θ = 1 to simplify the equation:
(sin 10)^2 + (sin 50)^2 + (sin 70)^2 = 3 - cos^2 10 + cos^2 80 + cos^2 20
= 3 - cos^2 10 + (1 - sin^2 80) + (1 - sin^2 20)
= 3 - cos^2 10 + 1 - sin^2 80 + 1 - sin^2 20
8. Simplify further:
(sin 10)^2 + (sin 50)^2 + (sin 70)^2 = 3 - cos^2 10 + 1 - sin^2 80 + 1 - sin^2 20
= 3 - cos^2 10 - cos^2 10 + 2
= -2cos^2 10 + 5
9. We can express cos^2θ in terms of sin^2θ using the identity cos^2θ = 1 - sin^2θ:
(sin 10)^2 + (sin 50)^2 + (sin 70)^2 = -2(1 - sin^2 10) + 5
= -2 + 2sin^2 10 + 5
= 2sin^2 10 + 3
From this equation, we can see that it differs from the desired equation 3/2.
Therefore, the original equation (sin 10)^2 + (sin 50)^2 + (sin 70)^2 = 3/2 is incorrect.