<PRE><B>Question 894869</B>
4 = 2*cos(pi/6*x - 2*pi/3) + 4


subtract 4 from both sides of the equation to get:


0 = 2*cos(pi/6*x - 2*pi/3)


this is the same as:


2*cos(pi/6*x - 2*pi/3) = 0


since cos(pi/6*x - 2*pi/3) = cos(pi/6*x)*cos(2*pi/3) + sin(pi/6*x)*sin(2*pi/3), and since cos(2*pi/3) = -1/2 and since sin(2*pi/3) = sqrt(3)/2, then you have:


cos(pi/6*x - 2*pi/3) = cos(pi/6*x)*(-1/2) + sin(pi/6*x)*(sqrt(3)/2)


plug this back into your original equation of 2*cos(pi/6*x - 2*pi/3) = 0 and you get:


2*(cos(pi/6*x)*-(1/2) + sin(pi/6*x)*(sqrt(3)/2) = 0


simplify by distributing the multiplication to get:


-cos(pi/6*x) + sqrt(3)*sin(pi/6*x) = 0


add cos(pi/6*x) to both sides of this equaiton to get:


sqrt(3)*sin(pi/6*x) = cos(pi/6*x)


divide both sides of this equation by cos(pi/6*x) to get:


sqrt(3)*sin(pi/6*x) / cos(pi/6*x) = 1


divide both sides of this equation by sqrt(3) and you get:


sin(pi/6*x) / cos(pi/6*x) = 1/sqrt(3)


since sin/cos = tan, you get:


tan(pi/6*x) = 1/sqrt(3)


solve for pi/6*x to get pi/6*x = arctan(1/sqrt(3))


solve for x to get x = 6*arctan(1/sqrt(3))/pi to get x = 1


that&#39;s your solution.


x is equal to 1.


plug that back into your original equation to see if that&#39;s true.


4 = 2*cos(pi/6*x - 2*pi/3) + 4 becomes:


4 = 2*cos(pi/6 - 2pi/3) + 4


simplify this to get:


4 = 2*cos(-pi/2) + 4 which becomes:


4 = 0 + 4 which becomes 4 = 4


this confirms the solution is correct.


if you solved this graphically, you would graph:


y = 4 and y = 2*cos(pi/6*x - 2*pi/3) + 4


the intersection of the 2 equations would be your solution.


that graph is shown below:


&lt;img src = &quot;http://theo.x10hosting.com/2014/082502.jpg&quot; alt=&quot;$$$&quot; &lt;/&gt;


to solve this equation you used the trigonometric identity of:


cos(a-b) = cos(a)*cos(b)+sin(a)*sin(b)


you also used the fact that 2pi/3 radians is equivalent to 120 degrees and cosine of 120 degrees is equivalent to minus cosine of 60 degrees which is equal to -1/2.


you also used the fact that 2pi/3 radians is equivalent to 120 degrees and sine of 120 degrees is equivalent to sin of 60 degrees which is equal to sqrt(3)/2.


120 degrees is in the second quadrant.


the reference angle for 120 degrees is 180 - 120 = 60 degrees.


that&#39;s the equivalent angle in the first quadrant.


in the second quadrant, sine is positive and cosine is negative.











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