Why Does tan x Have This Domain?

In summary, the domain of tan x is not all real numbers, but rather a set of numbers that make the denominator of the function equal to zero. This set does not reflect the defined values of the function, but rather the undefined ones. The range of the tangent function is all real numbers.
  • #1
shihab-kol
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8
I found in a book that the domain of tan x was {(2n+1)π/2 , n∈I}
The graph however shows that for every value of x , the function takes on a value .So, why is the domain like this?
 
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  • #2
shihab-kol said:
I found in a book that the domain of tan x was {(2n+1)π/2 , n∈I}
The graph however shows that for every value of x , the function takes on a value .So, why is the domain like this?
##\tan(x) = \frac{\sin(x)}{\cos(x)}## so whenever ##\cos(x) =0## then ##\tan(x)## isn't defined.
 
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  • #3
shihab-kol said:
I found in a book that the domain of tan x was {(2n+1)π/2 , n∈I}
That set is the complement of the domain of the tan function -- the set of x values where tan x is undefined.
 
  • #4
So that set does not reflect the defined values of the function,just the undefined ones. Right?
Correct me if I am wrong.
 
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  • #5
shihab-kol said:
So that set does not reflect the defined values of the function,just the undefined ones. Right?
Correct me if I am wrong.
The set you wrote in post #1 is not the defined values of the function. The range of the tangent function (set of all output values) is all real numbers. That set in post #1 lists (by a formula) all of the numbers that are not in the domain of tan(x).
 
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1. Why is the domain of tan x limited?

The domain of tan x is limited because the tangent function is undefined at certain values of x. Specifically, tan x is undefined at odd multiples of pi/2, such as pi/2, 3pi/2, 5pi/2, etc. This is because at these values, the denominator of the tangent function becomes zero, which is undefined.

2. Can the domain of tan x be extended?

No, the domain of tan x cannot be extended. The definition of the tangent function is limited to the values of x where the function is defined, and it cannot be extended to include undefined values. However, the range of tan x can be extended to include all real numbers.

3. Why is the domain of tan x different from other trigonometric functions?

The domain of tan x is different from other trigonometric functions because the tangent function is a ratio of sine and cosine, and both of these functions have limited domains. Sine and cosine are also periodic, which means they repeat their values after a certain interval, creating gaps in the domain of tangent.

4. What happens to the graph of tan x when x approaches the values in its domain?

As x approaches the values in the domain of tan x, the graph of the function will approach infinity or negative infinity. This is because the tangent function has vertical asymptotes at the values where it is undefined, causing the graph to approach these values but never reach them.

5. How does the limited domain of tan x affect its applications?

The limited domain of tan x can affect its applications in certain situations where the values of x may approach the undefined values. This can lead to errors in calculations or incorrect interpretations of data. It is important to be aware of the domain of tan x and its limitations when using it in applications.

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