Antianna W.

asked • 04/21/16

cos(7x)+cos(9x)=cos(x)

find all solutions of the equation using the sum-to-product formula

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Mark M. answered • 04/21/16

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cos 7x + cos 9x = cos x
2 cos ((7x+9x)/2) cos ((7x-9x)/2) = cos x
2 cos 8x cos x = cos x
2 cos 8x = 1
cos 8x = 1/2
8x = π/3, or 8x = 5π/3
x = π/24, or x = 5π/24
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Kenneth S.

2 cos 8x cos x = cos x  You divided by cos x, which assumes cos x is not zero.
 
Should put all terms on one side, and factor; cos x = 0 implies that x = pi/2 or 3pi/2.
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cos 8x = 1/2
8x = π/3 +2kpi, or 8x = 5π/3 + 2kpi ... so when you divide by 8, you get numerous additional answers. in [0,2pi] by using various integer k values...
 
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04/21/16

Antianna W.

So what is the final answer for all x values?
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04/21/16

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