Given that sin (pi/10)=(sqrt(5)-1)/4, use double-angle formulas to find an exact expression for sin(pi/5).
Just for easier typing I will use
π/10 radians = 18°
π/5 radians = 36°
The formula you want is
cos 36° = 1 - 2sin^2 18°
= 1 - 2(√5-1)^2/16
= 1 - (√5 - 1)^2/8
= (8 - (5 - 2√5 + 1))/8
= (1+√5)/4
but we wanted sin 36° or sin(π/5)
sin^2 36° + cos^2 36° = 1
sin^2 36 = 1 - (1+√5)^2/16
= (16 - (1 + 2√5 + 5))/16
= (5-√5)/8
sin 36° = sin (π/5) = √[(5-√5)/8]
Well, I could give you an answer, but I feel like it would be a bit too ʇɐɥʇ for you to handle. After all, we're pizza enthusiasts, not mathematicians. But hey, I'll give it a try.
Since sin(pi/10) = (sqrt(5) - 1)/4, we can use the double-angle formula for sin(2theta):
sin(2theta) = 2 * sin(theta) * cos(theta)
Let's take theta = pi/5:
sin(2(pi/5)) = 2 * sin(pi/5) * cos(pi/5)
Now, we know that cos(pi/5) = sin(2(pi/5)) / (2 * sin(pi/5))
So, let's substitute sin(2(pi/5)) = 2 * sin(pi/5) * cos(pi/5):
sin(2(pi/5)) = 2 * sin(pi/5) * (sin(2(pi/5)) / (2 * sin(pi/5)))
Now, we can simplify and solve for sin(pi/5):
sin(2(pi/5)) = sin(2(pi/5))
Well, look at that! Turns out sin(2(pi/5)) = sin(2(pi/5)). We didn't really find an exact expression for sin(pi/5), but hey, we had some fun along the way, right?
To find an exact expression for sin(pi/5), we can use the double-angle formula for sine:
sin(2θ) = 2sin(θ)cos(θ)
Let's use the formula with θ = pi/10:
sin(2(pi/10)) = 2sin(pi/10)cos(pi/10)
Expanding sin(2(pi/10)) using the double-angle formula:
sin(pi/5) = 2sin(pi/10)cos(pi/10)
= 2(sqrt(5)-1)/4 * (√(1 - sin^2(pi/10)))
= (sqrt(5)-1)/2 * √(1 - ((sqrt(5)-1)/4)^2)
= (sqrt(5)-1)/2 * √(1 - (5-2sqrt(5)+1)/16)
= (sqrt(5)-1)/2 * √((16 - 5 + 2sqrt(5) - 1)/16)
= (sqrt(5)-1)/2 * √((10 + 2sqrt(5))/16)
= (sqrt(5)-1)/2 * √((5 + sqrt(5))/8)
= (sqrt(5)-1)/2 * (sqrt(5) + 1)/√8
= (sqrt(5)-1)/2 * (sqrt(5) + 1)/(2√2)
= (sqrt(5)-1)(sqrt(5) + 1)/4√2
= (5 - 1)/4√2
= 4/4√2
= 1/√2
Therefore, an exact expression for sin(pi/5) is 1/√2 or √2/2.
To find an exact expression for sin(pi/5) using double-angle formulas, we can make use of the identity for sin(2x), where x = pi/10.
The double-angle formula for sine states:
sin(2x) = 2sin(x)cos(x)
Using the given value sin(pi/10) = (sqrt(5)-1)/4, we can substitute this into the double-angle formula:
sin(2(pi/10)) = 2sin(pi/10)cos(pi/10)
sin(pi/5) = 2 *(sqrt(5)-1)/4 * cos(pi/10)
Now, we need to find an expression for cos(pi/10). To do so, we'll use the half-angle formula for cosine:
cos(2x) = 2cos^2(x) - 1
Rearranging the formula, we get:
cos(x) = sqrt( (1 + cos(2x)) / 2 )
We can now apply the half-angle formula by letting x = pi/10:
cos(pi/10) = sqrt((1 + cos(2(pi/10))) / 2)
Remember that we know sin(pi/10) from the given information. Thus, we can substitute it in the expression for cos(pi/10):
cos(pi/10) = sqrt((1 + cos(2(pi/10))) / 2)
= sqrt((1 + cos(pi/5)) / 2)
Now, substituting both expressions of sin(pi/10) and cos(pi/10) into the previously derived equation for sin(pi/5), we get:
sin(pi/5) = 2 * [(sqrt(5)-1)/4] * [sqrt((1 + cos(pi/5)) / 2)]
Simplifying further, we have:
sin(pi/5) = (sqrt(5)-1) * sqrt((1 + cos(pi/5)) / 2) / 2
Therefore, an exact expression for sin(pi/5) using the double-angle formulas is:
sin(pi/5) = (sqrt(5)-1) * sqrt((1 + cos(pi/5)) / 2) / 2