Question

solve 2tan(3x) = 7 over domain all real numbers
not sure if answer is okay and the graph. I will upload a photo

Answer 1

no I do not have a graph at the moment, just the question

Answer 2

Trigonometry in itself is a relationship of equivalence, this implying that each trigonometric entity actually stems from the other, and this forms the base of the trigonometry.
When we find a variable and an equal sign, we know we a are in front of a trigonometrical equation, what does this imply?
it implies that we trying to find the value of the variable, relying on the reciprocal trigonometric rules and conveniently isoltaing the variable, thus solving the problem.

Moving to the problem in question:
\[2\tan (3x)=7\]
Equations by themselves often present a problem which does not have to do with the operation itself but rather the variable and how it is presented, it is far more easy to work with \(\tan (u)\) instead of \(\tan (3x)\), this being because one has a number multplying which hides a little bit the whole "evident solution" thing.
When we are specified a domain, it is implied that we have to find a possible solution in some given interval, but since we are given \(x \in \mathbb{R}\) pretty much meaning we have no restrictions at all, and we can work with all the numbers.
Let's begin by efectuating the change of variable \(u=3x\), this will allow us to more simply operate this, and if we can solve for "u", we can later on solve for "x".
Replacing the information:
\[2\tan(u)=7\]
The first step to leaving the "u" isolated is to divide both sides by 2, which you should be able to perform, given the level of the excercise:
\[\tan(u)=\frac{ 7 }{ 2 }\]
So far o good, now, we perform the operation "arcotangent", which is the reciprocal of tangent, and notated as \(\tan ^{-1}(u)\), so let's now perform the operation:
\[\tan ^{-1}(\tan(u))=\tan ^{-1}(\frac{ 7 }{ 2 })\]
\[u=\tan ^{-1}(\frac{ 7 }{ 2 })\]
Look, we have isolated the "u" variable, as we wanted... and if we perform that on the calculator:
\[u \approx 1.2924\]
This is just an approximate value, you can use your calculator to consider all the decimals but in this post, I'll limit myself to four, should be enough.
Now that we know the value of "u", we can deconstruct the change of variable we did earlier, remember \(u=3x\)... We established it at the begining so we could operate it more simple, but let's now change all back:
\[3x=1.2924\]
And solve for "x":
\[x=\frac{ 1.2429 }{ 3 }\]
\[x \approx 0.4308\]
Now comes the part of the trigonometrical equations that bothers all, the part to study the frequency of the given trigonometric function, this being tangent, we know that the specific scenario for \(\tan(3x)\) is that every solution repeats every \(\frac{ 3 \pi }{ 2 }\) , this will allow us to find the second solution to this equation and finish off the excercise, we will look for that value by discarding the first solution found in the beginning, because it'd create unecessary problem to consider all the trigonometric circle to just find one solution, wo, let's efectuate it:
\[x_2=\frac{ 3 \pi}{ 2}-0.4308\]
\[x_2 \approx 4.2816\]

Notice that this gave us two solutions, just because this whole thing was in terms of tangent, if it were another trigonometric operation, we would have treated this a little different.

Answer 3

I think that we have these solutions:
\[\begin{gathered}
{x_1} = \frac{1}{3}\arctan \left( {\frac{7}{2}} \right) = 24.68 \hfill \\
{x_2} = \frac{1}{3}\arctan \left( {\frac{7}{2}} \right) + 180 = 204.68 \hfill \\
\end{gathered} \]

Answer 4

sorry, I have made an error, here are the right formulas
\[\begin{gathered}
3{x_1} = \arctan \left( {\frac{7}{2}} \right) = 74.05 \hfill \\
3{x_2} = \arctan \left( {\frac{7}{2}} \right) + 180 = 254.05 \hfill \\
\end{gathered} \]

Answer 5

3x1=arctan(72)=74.053x2=arctan(72)+180=254.05

Answer 6

no I do not have a graph at the moment, just the question

Answer 7

Please note that: since we have to consider this quantity \(3 x\) then the periodicity is 1/3 of \(180\)