How do you find all values of x in the interval [0, 2pi] in the equation $cos(x+pi/3)+cosx=0$ ?

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Answer 1 · 2016-03-14T09:51:12

In domain $[0,2pi]$ , solution is ${pi/3, (4pi)/3}$

We will use the identity $cos(x+y)=cosxcosy-sinxsiny$

Hence $cos(x+pi/3)+cosx=0$

or $cosxcos(pi/3)-sinxsin(pi/3)+cosx=0$

=or $cosx xx1/2-sinx xxsqrt3/2 +cosx=0$

or $sinx xxsqrt3/2=cosx xx3/2$

or $sinx/cosx=3/2xx2/sqrt3$

or $tanx=3/sqrt3=sqrt3$

or $x=pi/3$

But as $tan(pi+x)=tanx$ another solution is $pi+pi/3=(4pi)/3$ .

Hence, in domain $[0,2pi]$ , solution is ${pi/3, (4pi)/3}$

How do you find all values of x in the interval [0, 2pi] in the equation cos(x+pi/3)+cosx=0cos(x+pi/3)+cosx=0?