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If:
sin(3pi / 10) = (1 + squareroot 5) / 4
Find the exact value of cos(pi / 5)
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Thankyou very much. Of course, all help is appreciated.
this might be easier to handle if you think in degrees.
pi/5 radians = 36º and
3pi/10 rad = 54º
did you notice that they add up to 90º and did you know that the cos (any angle) = sin (90 -any angle) ?
so cos(36º)= sin(54º) = (1+√5)/4
BTW, cos 36º is exactly half the Golden Ratio, which would bring me to the Fibonacci numbers and other most fascination topics that I could spend hours ......
To find the exact value of cos(pi / 5), we can use the trigonometric identity:
cos^2(x) + sin^2(x) = 1
We have the value of sin(3pi / 10), so we can find cos(3pi / 10) using the identity above. Then, we can use another trigonometric identity:
cos(2x) = 1 - 2sin^2(x)
to find cos(6pi / 10), and finally use the identity:
cos(2x) = 2cos^2(x) - 1
to solve for cos(pi / 5).
Now, let's break down the steps:
Step 1: Find cos(3pi / 10)
Since we know sin(3pi / 10), we can use the identity cos^2(x) + sin^2(x) = 1 to find cos(3pi / 10):
cos^2(3pi / 10) = 1 - sin^2(3pi / 10)
cos^2(3pi / 10) = 1 - [(1 + sqrt(5)) / 4]^2
cos^2(3pi / 10) = 1 - (1 + sqrt(5))^2 / 16
cos^2(3pi / 10) = 1 - (1 + 2sqrt(5) + 5) / 16
cos^2(3pi / 10) = 1 - (6 + 2sqrt(5)) / 16
cos^2(3pi / 10) = (16 - 6 - 2sqrt(5)) / 16
cos^2(3pi / 10) = (10 - 2sqrt(5)) / 16
Taking the square root on both sides, we find:
cos(3pi / 10) = sqrt[(10 - 2sqrt(5)) / 16]
Step 2: Find cos(6pi / 10)
Using the identity cos(2x) = 1 - 2sin^2(x), we can substitute x = 3pi / 10 to find cos(6pi / 10):
cos(6pi / 10) = 1 - 2sin^2(3pi / 10)
cos(6pi / 10) = 1 - 2[(1 + sqrt(5)) / 4]^2
cos(6pi / 10) = 1 - 2(1 + sqrt(5))^2 / 16
cos(6pi / 10) = 1 - 2(1 + 2sqrt(5) + 5) / 16
cos(6pi / 10) = 1 - 2(6 + 2sqrt(5)) / 16
cos(6pi / 10) = 1 - (12 + 4sqrt(5)) / 16
cos(6pi / 10) = (16 - 12 - 4sqrt(5)) / 16
cos(6pi / 10) = (4 - 4sqrt(5)) / 16
cos(6pi / 10) = (1 - sqrt(5)) / 4
Step 3: Find cos(pi / 5)
Using the identity cos(2x) = 2cos^2(x) - 1, we can substitute x = 3pi / 10 to find cos(pi / 5):
cos(pi / 5) = 2cos^2(3pi / 10) - 1
cos(pi / 5) = 2[(10 - 2sqrt(5)) / 16]^2 - 1
cos(pi / 5) = 2(10 - 2sqrt(5))^2 / 16 - 1
cos(pi / 5) = (10 - 2sqrt(5))^2 / 8 - 1
cos(pi / 5) = [(10 - 2sqrt(5))^2 - 8] / 8
cos(pi / 5) = [(100 - 40sqrt(5) + 20) - 8] / 8
cos(pi / 5) = (112 - 40sqrt(5)) / 8
cos(pi / 5) = 14 - 5sqrt(5)
Therefore, the exact value of cos(pi / 5) is 14 - 5sqrt(5).