Given: cos x + cos (3x) + cos (5x). what is x?

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neela | High School Teacher | (Level 3) Valedictorian

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cosx+cos3x+cos5x is an expression which could be simplified. It is not an equation. However, the expression  is equated to 0 and the solution for x is given in the following para.

Cosx+cos3x+cos5x=0

Cosx+{cos3x+cos5x}=0

cosx+2cos[(3x+5x)/2]cos[(5x-3x)/2] = 0

cosx+2cos4x*cosx = 0

cosx{1+2cos4x} = 0

cosx=0  or 1+2cos4x = 0

Cosx =0, for x=90 and 270 degrees

1+2cos4x = 0 gives:

cos4x = -1/2.

4x = 120  or 4x = 240 .  General solution  4x= (n*360 +or- 120) degree. So,

x = n*90+30 or x=n*90-30 degree.

The angles,30 or 120  or 150  or 210  300 and 330 degrees  hold good  in 0 to 360 dgrees.(p/6,4p/6, 5p/3,7p/6,10p/6 and 11p/6 radians)

 

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