Why do sin, cos, tan, and their reciprocals have different domains?

[name_ask17]

Homework Statement

Ok, this is not really a problem, but I need help on understanding the basics of sin, cos, tan, and their inverses.

i was looking at http://www.analyzemath.com/DomainRange/domain_range_functions.html and it was saying that the domain for sin and cos is (-inf , + inf)
and then for tan it is All real numbers
except pi/1 + n*Pi
but then for csc, it is All real numbers
except n*Pi

Can you explain why? I think I'm having trouble with figuring out how to find domains and I want to understand this before I start calculus. Please explain.

 

[SteamKing]

You might want to check the expression for x where tan x goes to +inf or -inf.

 

[Hells]

You need to try a little yourself before asking, but I'll aid you this once..

Tan(x) = Sin(x)/Cos(x)

csc(x) = 1/Sin(x)

What's the rule for dividing?

 

[Mark44]

Homework Statement

Ok, this is not really a problem, but I need help on understanding the basics of sin, cos, tan, and their inverses.

i was looking at http://www.analyzemath.com/DomainRange/domain_range_functions.html and it was saying that the domain for sin and cos is (-inf , + inf)
and then for tan it is All real numbers
except pi/1 + n*Pi

I don't know what to make of "pi/1 + n*Pi ". The domain for the tangent function is all real numbers x, such that x ≠ (2n + 1)∏/2, where n is an integer. IOW, all reals except odd multiples of ∏/2.

 

[eumyang]

and then for tan it is All real numbers
except pi/1 + n*Pi

Correction: this should be
$$\frac{\pi}{2} + n\pi$$
. And you have to specify what n can equal, as Mark44 said.

 

[AJKing]

Hi Name_Ask,

I'd like you to go to http://www.touchtrigonometry.org/" and play around with it a little bit.

While you're there, make sure to do the following:

• Look at the bottom left of the screen where you see the tig. function names and a value beside each.
  
• Turn them all off by clicking on them.
  
• Turn one on at a time.
  
• Take notice of how often its pattern repeats, and when it starts.
  
• Examine all the "x" values it can hold and the ones that are impossible.
  
• Why are some of these Tig values impossible?
  
• Click the active graph at any time to "Pause" your mouse, and look at what the line does on the Trig Circle to the left.
  
• Compare what you see with your knowledge of what happens when a number is divided by 0.
  
• Repeat with a new trig function.