You can put this solution on YOUR website! As is often the case, there are several ways to solve this. Most, if not all, of them will involve getting the arguments of tan to be the same.
One solution is to rewrite tan(6x) as tan(3x+3x) and then use the tan(A+B) formula:
tan(6x) = tan(3x)
tan(3x+3x) = tan(3x)
Using this with A = 3x and B = 3x on tan(3x+3x) we get:
which simplifies to:
Now that all the tan's have the same argument, 3x, we will solve for tan(3x). First we will eliminate the fraction by multiplying each side by :
If this looks like a difficult equation to solve, it may help to use a temporary variable:
Let q = tan(3x)
Substituting this into the equation we get:
To solve this kind of equation we want one side to be zero. Subtracting 2q from each side we get:
Now we factor. As always with factoring start with the GCF (q or -q):
Now from the Zero Product property we know that this product can be zero only if one (or more) of the factors is zero. So:
-q = 0 or
From the first equation we get q = 0. And since it is impossible to get zero by adding 1 to a perfect square (think about it), there is no way for the second factor to be zero. So the only solution is q = 0.
Now we can replace our temporary variable:
tan(3x) = 0
This tells us that 3x must be an angle whose tan is 0. Since tan(0) = 0 and since the period of tan is we get: (where "n" is any integer)
Last of all we divide by 3:
or
This is a general solution which expresses all of the infinite number of x's that fit tan(6x) = tan(3x). Specific individual solutions can be found by using various integer values for "n" and simplifying. For example, with n = 2 we get:
or
(