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
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up vote
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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
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
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
2,170631
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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BMNbl k,xHN,QB9dXmf50r4dRF8edO
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