What is tan3x(tanx-1)=0 ?

Question

Solve the equation for 0 ≤ x < 2π

tan3x(tanx-1)=0

I am trying to help out a friend who is taking pre-calculus and I haven't worked these out in soo long. Thanks for any help you may provide!

Answer 1

Note that this is just a product of functions equal to 0, so we just equate each one to 0.

tan(3x) = 0 and tanx - 1 = 0

We have to work on these individually. Let's work on tanx - 1 = 0 first.

tanx - 1 = 0
tanx = 1

Here is where you ask yourself: On that unit circle you were asked to study, where is tan equal to 1? The answer to that is in the first and third quadrant: pi/4 and 5pi/4

For tan(3x) = 0 (which is a bit tricker), you ask yourself again: Where is tan equal to 0? The answer is when sine is equal to 0, which is at 0 and pi

Therefore, 3x = 0, pi, ?, ?, ?, ?
I put question marks there because the 3x implies you are effectively tripling the number of solutions you have. What you have to do is subsequently add 2pi to each solution, and then add 2pi to each new solution. Visually I'll show you below.

3x = 0, pi, 2pi, 3pi, 4pi, 5pi

Now, divide each solution by 3.

x = 0, pi/3, 2pi/3, pi, 4pi/3, 5pi/3

Therefore, all the solutions are:

x = pi/4, 5pi/4, 0, pi/3, 2pi/3, pi, 4pi/3, 5pi/3

A note about examples like sin(Ax), cos(Ax), tan(Ax):

Whenever you have sin(Ax) where A is a constant, you'll be having A times more solutions. So if you get a solution for sin(2x), you'll have twice as many solutions as normal.

For instance, if

sin(2x) = 1/2, then
2x = pi/6, 5pi/6, ?, ?

And then you'd continue writing solutions in the following pattern

2x = pi/6, 5pi6, pi/6 + 2pi, 5pi/6 + 2pi

Notice that in this case there are 4 total solutions, whereas in the tan3x case there were 6. That's what I mean when i say "A times more solutions" for sin(Ax).

2x = pi/6, 5pi/6, 13pi/6, 17pi/6
x = pi/12, 5pi/12, 13pi/12, 17pi/12

Note that all solutions are within the boundary 0 <= x < 2pi

Answer 2

That means

tan 3x = 0 or tan x -1 =0

tan x - 1= 0 means tan x = 1

x = pi/4 or pi+pi/4( 1st and 3rd quadrant tan is positive)

tan 3x = means x = npi/2

put pi = 0 to 5 and get 5 vavlues < 2 * pi

Answer 3

The equation is
tan 3x (tan x - 1) = 0

Thus, by the zero product property,
tan 3x = 0 or tan x - 1 = 0

Then
3x = arctan 0 or tan x = 1

Thus,
x = arctan 0/3 or x = arctan 1

Therefore, the values for x are
x = arctan 0/3 or x = π/4, 5π/4

^_^

Answer 4

Either tan(3x) = 0 OR tan(x)-1 = 0

So x = Pi*k/3 (for some integer k) OR x = Pi/4 or 5Pi/4

The only values of k that give x between 0 and 2Pi are 0,1,2,3,4 & 5

So x has to be one of (0,Pi/4,Pi/3,2Pi/3,Pi,4Pi/3,5Pi/4,5Pi/3)

Answer 5

tan3x(tanx-1)=0
tan(x-1)=0
tanx=1
when does tangent equal one??

Pi/4

Answer 6

Great point, but I'm not 100%

Answer 7

Jeff Johnson and Jamie Hopkins asked the same question. You should read the answers side by side.

Answer 8

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