What is general solution of \frac{\cos 5x \cos 3x-\sin 3x

David Young

David Young

Answered question

2021-12-31

What is general solution of cos5xcos3xsin3xsinxcos2x=1
1) kπ3
2) kπ2
3) 2kπ5
4) 2kπ3
The numerator of the fraction is cos(5x+3x). so I should find general solution of cos8x=cos2x. I'm not sure how to do it, I can write cos8x in term of cos2x:
cos8x=2cos24x1=2(2cos22x1)21
After substituting it in the equation and using cos2x=t we have degree four equation

Answer & Explanation

jean2098

jean2098

Beginner2022-01-01Added 38 answers

0=cos5xcos3xsin3xsinxcos2x
=cos5xcos3xsin3xsinxcos(5x3x)
=sin3xsin5xsin3xsinx
2sin23xcos2x=0
Note that cos2x0 has to be included
sin3x=0
William Appel

William Appel

Beginner2022-01-02Added 44 answers

cos(5x)cos(3x)sin(3x)sin(x)=cos(2x)
12cos(2x)+cos(8x)12cos(2x)cos(4x)=cos(2x)
12cos(4x)+cos(8x)cos(2x)=0
cos(8x)+cos(4x)2cos(2x)=0
2cos(2x)cos(6x)2cos(2x)=0
cos(2x)[cos(6x)1]=0
cos(2x)=0x=±π4+kπ
discarded because this solution will make zero the denominator of the original equation
cos(6x)=1x=kπ3

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