Expert: Paul Klarreich - 10/5/2009

Question
Question: Explain why the period of the y = tan(theta) and y = cot(theta) is only 180 degrees instead of 360 degrees, like the four other trigonometric functions.

What I thought about:
First tangent = y/x
y represents sine
x represents cosign

In the quadrants of the of a graph, sine is positive on the 1st/2nd quadrant, while being negative on the 3rd/ 4th quadrant.

For cosign, positive on the 1st/4th, negative on the 2nd/3rd quadrants.

Might a series of positive,negative,positive pattern relating to 180 degrees turn? Starting over the pattern every 180 degrees, but i still don't get why? Can you help me on this questions please? Thanks

Answer
Questioner: Walton
Country: United States
Category: Advanced Math
Question: Question: Explain why the period of the y = tan(theta) and y = cot(theta) is only 180 degrees instead of 360 degrees, like the four other trigonometric functions.

What I thought about:
First tangent = y/x
y represents sine
x represents cosign

In the quadrants of the of a graph, sine is positive on the 1st/2nd quadrant, while being negative on the 3rd/ 4th quadrant.

For cosine, positive on the 1st/4th, negative on the 2nd/3rd quadrants.

Might a series of positive,negative,positive pattern relating to 180 degrees turn? Starting over the pattern every 180 degrees, but i still don't get why? Can you help me on this questions please? Thanks

Hi, Walton,

I believe you are over-thinking this.  (I have a friend who accuses me of that, too.)

Best is to go back to definitions:

A function f(x) is periodic with period P if for every x,  f(x + P) = f(x).

Now you can show, for example:

sin(x + 2pi) = sin x

(Btw, degrees are for children.  You are a grownup now, if you are in precalc.)

And you do it by expanding;

sin(A + B) = sin A cos B + cos A sin B

sin(x + 2pi) = sin x cos 2pi + cos x sin 2pi

But cos 2pi = 1, and  sin 2pi = 0, so that is:

sin(x + 2pi) = sin x (1) + cos x (0) = sin x.

....................................

Now use:
             tan A + tan B  
tan(A + B) = ------------------
            1 - tan A tan B

Apply it to:
              tan x + tan pi  
tan(x + pi) = ------------------
             1 - tan x tan pi

And  tan pi = 0, so:

              tan x + 0
tan(x + pi) = -------------- = tan x
             1 - tan x (0)

thus showing the period to be pi.
....................................
You say you're not satisfied?  You say you want more for your money?  Tell you what I'm going to do:  (Ask your grandfather about that wording)

We have tan t = y/x, the definition.  (t = theta)

Suppose that  (x1,y1) is a point on the terminal side of angle t.  Now add pi to t.  Where is the corresponding point?  It is on the opposite side of the origin, and it has coordinates (-x1,-y1).

[comment:  "-x1" means "the opposite of x1", and may be positive or negative or zero.]

Then  tan(t + pi) = -y1/(-x1) = y1/x1 = tan t.