Solving a trig. equation with tan and finding the period

V

Velvet

New member
Joined
Nov 18, 2005
Messages
7
Hello,

I have a couple of questions in analytic trigonometry, any help is greatly appreciated!

Let's consider the following equation:
3tan^2 X - 1= 0
tan^2X = 1/3
tanX = + or - 1/square root of 3

Up to there, I understand. But then the textbook says that "tan X has a period of Pi". Why is that? Does tanX always have a period of Pi, or is only in this specific equation?

I thought that to find the period, we had to solve for period = 2Pi/b. Does it only work when we have an equation with sin or cos?

Also, how do we know that the answer is supposed to be x = Pi/6 + nPi and 5Pi/6 + nPi. WITHOUT using the calculator?

Thank you for any light you can cast on these questions!

V.
 
f(x) = tan(x) ALWAYS has a period of pi. The 2*pi thing is for functions with a period of 2*pi. The period is a fundamental property of the six basic trig functions. It is something you should memorize. Unless the problem specifically states a Domain, you should include ALL possible solutions.
 
Yes, tan(x) alwas has a period of pi.
If you look at the unit circle and follow a triangle with legs x & y as it swings around
quad 1 tan(x) = y/x
quad 2 tan(x) = x/-y = -x/y
quad 3 tan(x) = -y/-x = y/x
quad 4 tan(x) = -x/y

so it repeats every pi turns of r and the period is pi.

I assume 2Pi/b comes from sin(bx)?
If you follow the triangle around quads for sin it goes
y/1, x/1, -y/1, -x/1
and it repeats every 2pi turns of r so the period is 2pi. Cos does about the same thing, period 2pi.

Consider: If tan(x)=1/sqrt(3) then the triangle has legs of 1 & sqrt(3) and a hypotenuse of 2. That should bring to mind an old friend the 1,2,sqrt(3) (30°,60°,90°) triangle in which the 30° angle has that tan. Changing 30° to radians gives 30*pi/180 = pi/6
 
Thank you very much, that was very helpful!

I suppose that I should memorize that a triangle with sides = to 1,2 and sqrt(3) has angles = to 30, 60 and 90 degrees, and that its tan is = to 1/sqrt(3) then?

V.
 
Seems like it shows up in half the problems so YES memorize the triangle. You don't have to memorize the trig functions. If you suspect it applies, you can scribble the triangle and use sin = o/h, cos = a/h and tan = o/a (opposite, adjacent & hypotenuse)
---------------------
Gene
 
You must log in or register to reply here.
Top