Thread: Combination of Positive Integers

Please help. I'm taking an honors geometry class and my teacher gave us this problem as extra credit. I unfortunately am having a problem understanding exactly where to go with this one. I understand combination of positive integers but when the cos is thrown in I am confused.

Thanks

---Susan

Express cos(3m) as a combination of positive integer powers of cos(m). Use this expression to obtain a cubic polynomial p(x) with rational number coefficients such that x = cos(m) is a root. Graph the resulting polynomial for m = 20 degrees that displays all three of the roots.

Originally Posted by atvfan

Please help. I'm taking an honors geometry class and my teacher gave us this problem as extra credit. I unfortunately am having a problem understanding exactly where to go with this one. I understand combination of positive integers but when the cos is thrown in I am confused.

Thanks

---Susan

Express cos(3m) as a combination of positive integer powers of cos(m). Use this expression to obtain a cubic polynomial p(x) with rational number coefficients such that x = cos(m) is a root. Graph the resulting polynomial for m = 20 degrees that displays all three of the roots.

identities you'll need ...

1. sum identity for cosine



2. double angle identities for cosine and sine





3. pythagorean identity




... start with the first identity, and the fact that

cos2m= 2cos^2 m -1
sin2m=2sin(m)cos(m)
cos (A+B)= cosAcosB-sinAsinB

Cos(2m+m)=cos3m=(2cos^2m-1)cos(m) - sin(m)(2sin(m)cos(m))
=2cos^3(m) - cos(m) - 2(sin^2m)cos(m)
=2cos^3(m) - cos(m) - 2(1-cos^2(m))cos(m)
=2cos^3(m) - cos(m) -2cos(m) +2cos^2(m)
=4cos^3(m)-3cos(m)

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