2 + cos ⁡ ( 2 x ) = 3 cos ⁡ ( x )Find all values of X in the interval [ 0 , 2 π ] that satisfythe equation

Question

2  +  cos  ⁡  (  2  x  )  =  3  cos  ⁡  (  x  )Find all values of X in the interval   [  0  ,  2  π  ] that satisfythe equation

Ashlynn Stephens · Accepted Answer

from half angle formula.      cos          2        ⁡  x  =  1      /    2  +  1      /    2  ∗      cos          2        ⁡  x      cos          2        ⁡  x  −  1      /    2  =  1      /    2  ∗      cos          2        ⁡  x  2      cos          2        ⁡  x  −  1  =      cos          2        ⁡  xsub this in for       cos          2        ⁡  x.  2  +  2      cos          2        ⁡  x  −  1  =  3  cos  ⁡  x  2      cos          2        ⁡  x  −  3  cos  ⁡  x  +  1  =  0let   cos  ⁡  x  =  u  2      u          2        −  3  u  +  1  =  0u = 1/2, u = 1therefore  cos  ⁡  x  =  1      /    2     o  r     cos  ⁡  x  =  1  x  =  π      /    3  ,  x  =  5  π      /    3  ,  x  =  0  ,  x  =  2  π

daniellex0x0xto · Accepted Answer

cos  ⁡  (  2  x  )  =      cos          2        ⁡  x  −      sin          2        ⁡  x  2  +  cos  ⁡  (  2  x  )  =  3  cos  ⁡  (  x  )  2  +      cos          2        ⁡  x  −      sin          2        ⁡  x  =  3  cos  ⁡  x  2  +      cos          2        ⁡  x  −  (  1  −      cos          2        ⁡  x  )  =  3  cos  ⁡  x  2  −  1  +  2      cos          2        ⁡  x  =  3  cos  ⁡  x  2      cos          2        ⁡  x  −  3  cos  ⁡  x  +  1  =  0  (  2  cos  ⁡  x  −  1  )  (  cos  ⁡  x  −  1  )  =  0  2  cos  ⁡  x  −  1  =  0          cos  ⁡  x  −  1  =  0  2  cos  ⁡  x  =  1             cos  ⁡  x  =  1  cos  ⁡  x  =  1      /    2            x  =  arccos  ⁡  1  x  =  arccos  ⁡  1      /    2  x  =  π      /    3  ,  5  π      /    3  ,  x  =  0  ,  2  π

2+cos(2x)=3cos(x) Find all values of X in the interval [0, 2pi] that satisfythe equation

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