Help with domain of a tangent equation?

Question

bruno102 · Answer

What is the domain of y = tan(x) + 2?

jim_thompson5910 · Answer

keep in mind that 

tan(x) = sin(x)/cos(x)

jim_thompson5910 · Answer

just by definition

when is cos(x) equal to 0?

bruno102 · Answer

i'm not sure?

jim_thompson5910 · Answer

do you have a unit circle?

bruno102 · Answer

I know what it is

jim_thompson5910 · Answer

which points on the unit circle have an x coordinate of 0?

bruno102 · Answer

$$\pi/2$$

jim_thompson5910 · Answer

where else?

bruno102 · Answer

$$3\pi/2$$

jim_thompson5910 · Answer

so in general, how do we classify all of the solutions to cos(x) = 0 ?

mathmale · Answer

So...going back to the definition of the tangent function, what would happen if the angle were pi/2?  3pi/2?  Would the cosine function be defined or not?  Would the tangent function be defined or not?  Answering these questions correctly will help you state the domain of the tangent function.

bruno102 · Answer

I don't know

jim_thompson5910 · Answer

if we start with pi/2

how do we get to 3pi/2? what do we add onto pi/2 to get there?

bruno102 · Answer

3?

jim_thompson5910 · Answer

what is a full revolution of the circle? how many radians?

bruno102 · Answer

2?

jim_thompson5910 · Answer

close

jim_thompson5910 · Answer

not just 2 but 2____

bruno102 · Answer

2pi?

jim_thompson5910 · Answer

2pi yep

jim_thompson5910 · Answer

so half a revolution is 2pi/2 = pi radians

jim_thompson5910 · Answer

to go from pi/2 to 3pi/2, you add on pi


pi/2 + pi = pi/2 + 2pi/2

pi/2 + pi = (pi + 2pi)/2

pi/2 + pi = 3pi/2

jim_thompson5910 · Answer

adding on pi to 3pi/2 will then take you to

3pi/2 + pi = 3pi/2 + 2pi/2 
3pi/2 + pi = (3pi + 2pi)/2 
3pi/2 + pi = 5pi/2


And notice how cos(5pi/2) = 0

bruno102 · Answer

ok

jim_thompson5910 · Answer

so in general, to capture ALL of the solutions to cos(x) = 0, we would write 

$$\Large x = \frac{\pi}{2} + \pi*n$$

We start with one solution pi/2 and then we add on multiples of pi to get the other solutions. The 'n' is an integer.

bruno102 · Answer

ok i see

mathmale · Answer

Regarding your original post ("Help with domain of a tangent equation?"):

Since the tangent function is defined as$$\tan x =\frac{ \sin x }{ \cos x }$$the tangent function is defined wherever cos x is NOT zero.  

As discussed earlier, the cosine function is zero at pi/2 and at 3pi/2.  Thus, the tangent function is undefined at those two angles, but is definited everywhere in between those two angles.  This gives you a hint of what the domain of the tangent function is.

More often we recognize that the cosine is zero at -pi/2 and at +pi/2, which means that the tangent function is defined for angles between those two.

"Domain" here signifies sets of values of the angle at which the tangent function is definited.  So, as a starting point, we could start specifying the domain by writing (-pi/2,pi/2).  Is that all?  NO.  Why not?  Because the tangent function is periodic, with period pi.  Jim Thompson has handled this part of his explanation very well.

mathmale · Answer

A slightly different way to write out the domain would be to specify values that the angle may NOT have.  Using Jim's notation,

The domain of the tangent function tan x is$$x \neq\frac{ -\pi }{ 2 }\pm n \pi~OR ~ x \neq\frac{ \pi }{ 2 }\pm n \pi$$

mathmale · Answer

In this case it's easier to specify what angles (x-values) are NOT permitted than to specify which ones ARE permitted.