Expert: David Hemmer - 4/11/2007

Question
QUESTION: I learn a bit differently So I will need the "why we do this" answered along with the solving of the problem,if you don't mind.
Heres the problem:
Solve the equation

tan3x(tanx-1)=0

I know what I have to do. I'm just not sure how to do it.
first find the ref. angle
second find which quadrant I will be working in and then find the two answers.

The answers in the back of the book are (pi/3)+2npi,(2pi/3)+2npi. The answer is in radian form thus the pi.

Can you explain to me step by step how to go about solving this problem and also answering the question; why is 2pi added to the answers. I noticed this with other problems but I'm not sure of it's purpose.
If you can answer my questions I will be very grateful.
Thank You
cara
ANSWER: I think you better send me the question above and be careful where your parenthesis are, because 0 should be one of the answers since tan0=0 and I don't see that on your list.

I can explain the 2Pi though. Remember sin(x+2Pi)=sinx and same for cos and tangent. Thus adding multiples of 2Pi   to the angle doesn't change the trig function. This should be obvious to you if you learned these using the unit circe, you are justgoing a full rotation around the circle. Feel free to double check this problem and send it in again.

---------- FOLLOW-UP ----------

QUESTION: Thanks for your reply. We haven't been taught the unit circle. I guess I'll have to learn it before the teacher decides to teach it. I realize every teacher has there own way of teaching. I don't particularly care for this teachers way.

npi/3,(pi/4)+npi are the correct solutions. In a different form it would be n60degrees and 45degrees plus n180.

Inorder to do this problem I have to first solve for x and that will give me my reference angle and depending on if I had to change tan to something else to find the ref. angle will tell me which quadrant I will be working in.
In other words I should have two answers because tan (for ex.) is only positive in quad. I and III.
Do you get what I'm talking about. I can e-mail you a problem that I have done already and correctly if it will help you better understand my problem.
thank you,
cara

Answer
tan3x(tanx-1)=0

When you multiply two things together and get 0 then one or the other must be zero. Thus you must have:

tan3x=0 or tanx=1.

Now tangent =0 only at multiples of Pi. Thus if 3x is a multiply of Pi then x is a multiply of Pi/3. Thus the set of solutions nPi/3 come from the tan3x being zero.

The other case is tanx=1. This happens for x=45 degrees, aka Pi/4. Also for x=5Pi/4. These two values are in the first and 3rd quadrants. Of course adding 2Pi does not change the value of these trig functions so you get:

x=Pi/4, 5Pi/4, 9Pi/4, etc.. These are the solutions Pi/4 + nPi.