PROBLEM OF THE WEEK 4
posted by zarcsman at
7:39 PM
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Weekly I'll pose a problem for you to think about and to submit solutions. The main goal of this is to develop skills in problem solving and encourage student’s interest in mathematics. Solutions are due within 1 week from the date of publication. The first 3 correct solvers will be awarded 5 points and their names will be posted here along with best the solution. Submit your solutions to mlzarco@engineer.com or posting your solutions directly here.
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3 Comments:
consider first the equation
cos(π/7) + cos(3π/7) + cos(5π/7) + cos(7π/7) + cos(9π/7) + cos(11π/7) + cos(13π/7) = 0
cos(π/7) + cos(3π/7) + cos(5π/7) + cos(9π/7) + cos(11π/7) + cos(13π/7) = -cos(π)
cos(π/7) + cos(3π/7) + cos(5π/7) + cos(9π/7) + cos(11π/7) + cos(13π/7) = 1
since cos(x) = cos(-x) = cos(2π-x)
cos(π/7) + cos(3π/7) + cos(5π/7) + cos[2π-(5π/7)] + cos[2π-(3π/7)] + cos[2π-(π/7)] = 1
cos(π/7) + cos(3π/7) + cos(5π/7) + cos(5π/7) + cos(3π/7) + cos(π/7) = 1
2[cos(π/7) + cos(3π/7) + cos(5π/7)] = 1
cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2
but cos(π-x) = -cos(x)
cos(5π/7) = cos[π-(2π/7)] = -cos(2π/7)
therefore:
cos(π/7) -cos(2π/7) + cos(3π/7) = 1/2
By
Anonymous, at 9:53 AM
walang bang solution na nagpu-prove talaga..as in ung traditional way of proving identities..?..(".)?..
By
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