Algbraic proof of compound periodicity of a function
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While studying trigonometric functions I keep encountering exercises of this sort:
$fleft(xright)=frac{tanleft(xright)}{1+sinleft(xright)}$. Show that the periodicity of $f(x)$ is $2pi $.
I have looked on the Internet and the most I could find was stating that since the periodicity of $textbf{tan} =pi$ and the periodicity of $textbf{sin}=2pi$, then the compound periodicity is $2pi$. But why?
I'm learning Math all by myself, so I really can't afford to just memorize algorithms and formulas. Could you provide me with a thorough explanation of the process by which I can arrive at such conclusions? I would mostly value and algebraic proof.
functions trigonometry
edited 15 hours ago
Tianlalu
2,170631
asked 15 hours ago
Daniel Oscar
947
add a comment |
up vote
0
down vote
favorite
While studying trigonometric functions I keep encountering exercises of this sort:
$fleft(xright)=frac{tanleft(xright)}{1+sinleft(xright)}$. Show that the periodicity of $f(x)$ is $2pi $.
I have looked on the Internet and the most I could find was stating that since the periodicity of $textbf{tan} =pi$ and the periodicity of $textbf{sin}=2pi$, then the compound periodicity is $2pi$. But why?
I'm learning Math all by myself, so I really can't afford to just memorize algorithms and formulas. Could you provide me with a thorough explanation of the process by which I can arrive at such conclusions? I would mostly value and algebraic proof.
functions trigonometry
edited 15 hours ago
Tianlalu
2,170631
asked 15 hours ago
Daniel Oscar
947
Do you understand why $f(x) = f(x + 2pi)$, i.e. $2pi$ is a period of $f$?
– Mees de Vries
15 hours ago
Yes, I do. You get back to where you've started. I've a rather good intuitive notion of that concept. So maybe I'm not connecting the dots? But where?
– Daniel Oscar
15 hours ago
1
What do you mean "you get back to where you've started"? The periodicity $tau$ of a function $f$ is the least positive real such that $f(x + tau) = f(x)$ for all $x$. Showing that it is true for $tau = 2pi$ is half the work (but the easy part); now you have to show that it doesn't hold for a number smaller than $2pi$.
– Mees de Vries
15 hours ago
I understand, the period is a number that if you add it to any $x$ in $f(x)$ you obtain the same value in $y$.
– Daniel Oscar
15 hours ago
The periodicity of a compound function is when all functions provide the same value - this happens when usually after the longest periods. There are expressions that will result a shorter period but the "worst" is as described above.
– Moti
14 hours ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
While studying trigonometric functions I keep encountering exercises of this sort:
$fleft(xright)=frac{tanleft(xright)}{1+sinleft(xright)}$. Show that the periodicity of $f(x)$ is $2pi $.
I have looked on the Internet and the most I could find was stating that since the periodicity of $textbf{tan} =pi$ and the periodicity of $textbf{sin}=2pi$, then the compound periodicity is $2pi$. But why?
I'm learning Math all by myself, so I really can't afford to just memorize algorithms and formulas. Could you provide me with a thorough explanation of the process by which I can arrive at such conclusions? I would mostly value and algebraic proof.
functions trigonometry
edited 15 hours ago
Tianlalu
2,170631
asked 15 hours ago
Daniel Oscar
947
While studying trigonometric functions I keep encountering exercises of this sort:
$fleft(xright)=frac{tanleft(xright)}{1+sinleft(xright)}$. Show that the periodicity of $f(x)$ is $2pi $.
I have looked on the Internet and the most I could find was stating that since the periodicity of $textbf{tan} =pi$ and the periodicity of $textbf{sin}=2pi$, then the compound periodicity is $2pi$. But why?
I'm learning Math all by myself, so I really can't afford to just memorize algorithms and formulas. Could you provide me with a thorough explanation of the process by which I can arrive at such conclusions? I would mostly value and algebraic proof.
functions trigonometry
functions trigonometry
edited 15 hours ago
Tianlalu
2,170631
asked 15 hours ago
Daniel Oscar
947
edited 15 hours ago
Tianlalu
2,170631
asked 15 hours ago
Daniel Oscar
947
edited 15 hours ago
Tianlalu
2,170631
edited 15 hours ago
Tianlalu
2,170631
edited 15 hours ago
Tianlalu
2,170631
2,170631
asked 15 hours ago
Daniel Oscar
947
asked 15 hours ago
Daniel Oscar
947
asked 15 hours ago
Daniel Oscar
947
947
Do you understand why $f(x) = f(x + 2pi)$, i.e. $2pi$ is a period of $f$?
– Mees de Vries
15 hours ago
Yes, I do. You get back to where you've started. I've a rather good intuitive notion of that concept. So maybe I'm not connecting the dots? But where?
– Daniel Oscar
15 hours ago
1
What do you mean "you get back to where you've started"? The periodicity $tau$ of a function $f$ is the least positive real such that $f(x + tau) = f(x)$ for all $x$. Showing that it is true for $tau = 2pi$ is half the work (but the easy part); now you have to show that it doesn't hold for a number smaller than $2pi$.
– Mees de Vries
15 hours ago
I understand, the period is a number that if you add it to any $x$ in $f(x)$ you obtain the same value in $y$.
– Daniel Oscar
15 hours ago
The periodicity of a compound function is when all functions provide the same value - this happens when usually after the longest periods. There are expressions that will result a shorter period but the "worst" is as described above.
– Moti
14 hours ago
add a comment |
Do you understand why $f(x) = f(x + 2pi)$, i.e. $2pi$ is a period of $f$?
– Mees de Vries
15 hours ago
Yes, I do. You get back to where you've started. I've a rather good intuitive notion of that concept. So maybe I'm not connecting the dots? But where?
– Daniel Oscar
15 hours ago
1
What do you mean "you get back to where you've started"? The periodicity $tau$ of a function $f$ is the least positive real such that $f(x + tau) = f(x)$ for all $x$. Showing that it is true for $tau = 2pi$ is half the work (but the easy part); now you have to show that it doesn't hold for a number smaller than $2pi$.
– Mees de Vries
15 hours ago
I understand, the period is a number that if you add it to any $x$ in $f(x)$ you obtain the same value in $y$.
– Daniel Oscar
15 hours ago
The periodicity of a compound function is when all functions provide the same value - this happens when usually after the longest periods. There are expressions that will result a shorter period but the "worst" is as described above.
– Moti
14 hours ago
Do you understand why $f(x) = f(x + 2pi)$, i.e. $2pi$ is a period of $f$?
– Mees de Vries
15 hours ago
Do you understand why $f(x) = f(x + 2pi)$, i.e. $2pi$ is a period of $f$?
– Mees de Vries
15 hours ago
Yes, I do. You get back to where you've started. I've a rather good intuitive notion of that concept. So maybe I'm not connecting the dots? But where?
– Daniel Oscar
15 hours ago
Yes, I do. You get back to where you've started. I've a rather good intuitive notion of that concept. So maybe I'm not connecting the dots? But where?
– Daniel Oscar
15 hours ago
1
1
What do you mean "you get back to where you've started"? The periodicity $tau$ of a function $f$ is the least positive real such that $f(x + tau) = f(x)$ for all $x$. Showing that it is true for $tau = 2pi$ is half the work (but the easy part); now you have to show that it doesn't hold for a number smaller than $2pi$.
– Mees de Vries
15 hours ago
What do you mean "you get back to where you've started"? The periodicity $tau$ of a function $f$ is the least positive real such that $f(x + tau) = f(x)$ for all $x$. Showing that it is true for $tau = 2pi$ is half the work (but the easy part); now you have to show that it doesn't hold for a number smaller than $2pi$.
– Mees de Vries
15 hours ago
I understand, the period is a number that if you add it to any $x$ in $f(x)$ you obtain the same value in $y$.
– Daniel Oscar
15 hours ago
I understand, the period is a number that if you add it to any $x$ in $f(x)$ you obtain the same value in $y$.
– Daniel Oscar
15 hours ago
The periodicity of a compound function is when all functions provide the same value - this happens when usually after the longest periods. There are expressions that will result a shorter period but the "worst" is as described above.
– Moti
14 hours ago
The periodicity of a compound function is when all functions provide the same value - this happens when usually after the longest periods. There are expressions that will result a shorter period but the "worst" is as described above.
– Moti
14 hours ago
add a comment |
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Do you understand why $f(x) = f(x + 2pi)$, i.e. $2pi$ is a period of $f$?
– Mees de Vries
15 hours ago
Yes, I do. You get back to where you've started. I've a rather good intuitive notion of that concept. So maybe I'm not connecting the dots? But where?
– Daniel Oscar
15 hours ago
1
What do you mean "you get back to where you've started"? The periodicity $tau$ of a function $f$ is the least positive real such that $f(x + tau) = f(x)$ for all $x$. Showing that it is true for $tau = 2pi$ is half the work (but the easy part); now you have to show that it doesn't hold for a number smaller than $2pi$.
– Mees de Vries
15 hours ago
I understand, the period is a number that if you add it to any $x$ in $f(x)$ you obtain the same value in $y$.
– Daniel Oscar
15 hours ago
The periodicity of a compound function is when all functions provide the same value - this happens when usually after the longest periods. There are expressions that will result a shorter period but the "worst" is as described above.
– Moti
14 hours ago