Trig_N is the set of all real trig polynomic functions of highest order N, given as follows.
\(\displaystyle f(x) = a_0 $\displaystyle\sum\limits_{n=1}^N a_n*sin(n*x)$ + $\displaystyle\sum\limits_{n=1}^N b_n*cos(n*x)$\)
and, \(\displaystyle a_i, b_i a, is in R\)
I have to show that \(\displaystyle Trig_N(R)\) is a real vector space and that the functions \(\displaystyle sin^2(x)\) and \(\displaystyle cos^2(x)\) is in \(\displaystyle Trig_2(R).\)
Furthermore, this is given;
\(\displaystyle cos(A)*cos(B) = 1/2 [cos(A+B) + cos(A-B)]\)
\(\displaystyle sin(A)*sin(B) = 1/2 [-cos(A+B) + cos(A-B)]\)
\(\displaystyle f(x) = a_0 $\displaystyle\sum\limits_{n=1}^N a_n*sin(n*x)$ + $\displaystyle\sum\limits_{n=1}^N b_n*cos(n*x)$\)
and, \(\displaystyle a_i, b_i a, is in R\)
I have to show that \(\displaystyle Trig_N(R)\) is a real vector space and that the functions \(\displaystyle sin^2(x)\) and \(\displaystyle cos^2(x)\) is in \(\displaystyle Trig_2(R).\)
Furthermore, this is given;
\(\displaystyle cos(A)*cos(B) = 1/2 [cos(A+B) + cos(A-B)]\)
\(\displaystyle sin(A)*sin(B) = 1/2 [-cos(A+B) + cos(A-B)]\)
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