Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Find the general solution of the following equation:

cos 3x + cos x - cos 2x = 0










Answer

cos 3x + cos x - cos 2x = 0
rightwards double arrow         2 space cos open parentheses fraction numerator 3 straight x plus straight x over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator 3 straight x minus straight x over denominator 2 end fraction close parentheses minus cos space 2 straight x space equals space 0
rightwards double arrow         2 space cos 2 straight x space cosx space minus space cos space 2 straight x space equals 0
rightwards double arrow              cos space 2 straight x space left parenthesis 2 space cosx space minus space 1 right parenthesis space equals space 0
rightwards double arrow   either         cos 2x = 0                                      or    cos space straight x space equals space 1 half
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#6 {main}</pre>            or     cos space straight x space equals space cos straight pi over 3
rightwards double arrow    either            straight x space equals space left parenthesis 2 straight n plus 1 right parenthesis straight pi over 2 comma space straight n space element of space straight Z              or     straight x space equals space 2 straight pi space plus-or-minus space straight pi over 3 comma space space straight n space element of space straight Z
rightwards double arrow                       x = space space left parenthesis 2 straight n plus 1 right parenthesis space straight pi over 2                        or     straight x space equals space 2 straight pi plus-or-minus straight pi over 3 comma space straight n element of space straight Z

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