Trig Functions Periodicity: Which Function is Not Periodic?

[Saitama]

Homework Statement

Which one is not periodic?
(a)|sin 3x|+sin²x
(b)cos$\sqrt{x}$+cos²x
(c)cos 4x + tan²x
(d)cos 2x+sin x

Homework Equations

The Attempt at a Solution

I don't understand how to show whether the functions are periodic or not?

 

[SammyS]

Did you try graphing each function? Also, graph each term of each function.

 

[Saitama]

Did you try graphing each function? Also, graph each term of each function.

I don't want to use graphs for this question since i am not comfortable in adding them.
Any other method?

 

[Bohrok]

Look at the arguments of the trig functions; are any different than the others?

 

[SammyS]

What's the period of sin(3x)? What's the period of |sin(3x)| ?

What's the period of sin²(x) ?

Furthermore, regarding my previous post:

Even if you're not comfortable adding graphs, graphing each term for each problem may help you.​

 

[doppelganger]

For trigonometric functions, the period is generally expressed as a constant coefficient of the independent variable. Look through your choices for a term that does not contain a constant coefficient of the independent variable. Graph it to confirm that it is not periodic.

 

[NascentOxygen]

I don't want to use graphs for this question since i am not comfortable in adding them.
Any other method?

Not comfortable with graphing? Then that's precisely why you should do some graphing...and keep at it until you become perfectly comfortable with it.

That's the secret to becoming more proficient at graphing.

Here, try this: http://fooplot.com/"
Type a different equation in each colour box on the right.

I'll help you express your first equation: abs(sin(3x))+(sin(x))^2

 

[Saitama]

Look at the arguments of the trig functions; are any different than the others?

What do you mean by "arguments"?

What's the period of sin(3x)? What's the period of |sin(3x)| ?

What's the period of sin²(x) ?

Furthermore, regarding my previous post:

Even if you're not comfortable adding graphs, graphing each term for each problem may help you.​

How would i graph sin(3x)? And the period of sin²x is $\pi$.

For trigonometric functions, the period is generally expressed as a constant coefficient of the independent variable. Look through your choices for a term that does not contain a constant coefficient of the independent variable. Graph it to confirm that it is not periodic.

Sorry, i didn't get you. Please elaborate.

 

[Bohrok]

What do you mean by "arguments"?

The argument of sin(2x+5) is 2x+5, for cos(x+y) it's x+y, etc.

How would i graph sin(3x)? And the period of sin²x is $\pi$.

The 3 in sin(3x) affects the graph basically in only one way. What's the period of sin(3x) compared to sin(x)?
And it's correct, but how do you know the period of sin²x is $\pi$?

 

[Saitama]

The 3 in sin(3x) affects the graph basically in only one way. What's the period of sin(3x) compared to sin(x)?
And it's correct, but how do you know the period of sin²x is $\pi$?

Using wolfram alpha i found that the period of sin(3x) is $\frac{2\pi}{3}$ and sin² is $\pi$.

 

[SammyS]

How about |sin(3x)| ?

 

[I like Serena]

Hi Pranav-Arora!

Using wolfram alpha i found that the period of sin(3x) is $\frac{2\pi}{3}$ and sin² is $\pi$.

Hmm, the sine and cosine functions have a period of 2pi.
This means sin(x) = sin(x+2pi) = sin(x+4pi) = sin(x+6pi) = ...

So sin(3x) = sin(3x+2pi) = sin(3x+4pi) = ...

What period does this imply for x?

Similarly sin²x=(1 - cos(2x))/2
From this you can deduce its period...

 

[Saitama]

Hi I Like Serena!

Hmm, the sine and cosine functions have a period of 2pi.
This means sin(x) = sin(x+2pi) = sin(x+4pi) = sin(x+6pi) = ...

So sin(3x) = sin(3x+2pi) = sin(3x+4pi) = ...

What period does this imply for x?

Similarly sin²x=(1 - cos(2x))/2
From this you can deduce its period...

If i take this case: sin(3x)=sin(3x+2pi), then 3x gets canceled and i am left with 0=2pi. :(

How would i deduce the period of sin²x using the equation given by you?

 

[PeterO]

Homework Statement

Which one is not periodic?
(a)|sin 3x|+sin²x
(b)cos$\sqrt{x}$+cos²x
(c)cos 4x + tan²x
(d)cos 2x+sin x

Homework Equations

The Attempt at a Solution

I don't understand how to show whether the functions are periodic or not?

Post #13 makes it look like you don't recognise what the functions are?

 

[Saitama]

Post #13 makes it look like you don't recognise what the functions are?

I know what a function is.

 

[I like Serena]

If i take this case: sin(3x)=sin(3x+2pi), then 3x gets canceled and i am left with 0=2pi. :(

I was looking for sin(3x+2pi) = sin(3(x+2pi/3))
So if you add 2pi/3 to x, you'll get the same value for the sine function.
This means the period is 2pi/3.

How would i deduce the period of sin²x using the equation given by you?

Try again?

 

[PeterO]

I know what a function is.

So when you said,

If i take this case: sin(3x)=sin(3x+2pi), then 3x gets canceled and i am left with 0=2pi. :(

What how exactly were you going to cancel the 3x ?

 

[Saitama]

So when you said,

If i take this case: sin(3x)=sin(3x+2pi), then 3x gets canceled and i am left with 0=2pi. :(

What how exactly were you going to cancel the 3x ?

Because if sin x=sin y is given, can't we rewrite it as x=y?

 

[PeterO]

Because if sin x=sin y is given, can't we rewrite it as x=y?

NO. since sine is a periodic function, there is an infinite number of angles, which when put through the sine function give out the same value.

For example, try the following

sin 30
sin 150
sin 390
sin 750

There are lots of others that will give the same value.

 

[I like Serena]

Because if sin x=sin y is given, can't we rewrite it as x=y?

Nooooo.
Since sine has a period of 2pi, this means its value is the same whenever you add 2pi to its argument.
It's even worse, since sine takes on the same value twice in each period.

The proper way to rewrite sin x=sin y is:
x≡y (mod 2pi) or x≡pi-y (mod 2pi)

 

[Saitama]

NO. since sine is a periodic function, there is an infinite number of angles, which when put through the sine function give out the same value.

For example, try the following

sin 30
sin 150
sin 390
sin 750

There are lots of others that will give the same value.

They all give the same value, i.e 1/2

Nooooo.
Since sine has a period of 2pi, this means its value is the same whenever you add 2pi to its argument.
It's even worse, since sine takes on the same value twice in each period.

The proper way to rewrite sin x=sin y is:
x≡y (mod 2pi) or x≡pi-y (mod 2pi)

I don't know "mod" notation.

 

[I like Serena]

I don't know "mod" notation.

All right, so alternatively we can write:

x = y + 2 k pi or x = (pi - y) + 2 k pi, for any k that is a whole number.

(This is exactly what mod-notation means. )

 

[Saitama]

All right, so alternatively we can write:

x = y + 2 k pi or x = (pi - y) + 2 k pi, for any k that is a whole number.

(This is exactly what mod-notation means. )

How do you get "(pi-y)"?

 

[I like Serena]

How do you get "(pi-y)"?

You can see that if you look at the graph of the sine function.

The usual way to solve these equations, however, is by looking at the unit circle.
See http://en.wikipedia.org/wiki/Unit_circle" .

On the unit circle you can see that there are 2 angles, where sine takes on the same value.
That is, for a small angle t, the sine of t is the y-value on the right side of the unit circle belonging to this angle t.
But on the left side of the unit circle there is a second point with the same y-value.
The corresponding angle is pi minus the angle t.

Beyond that, adding a full period (2pi) of the unit circle, will also yield the same y-values again.

 

[PeterO]

They all give the same value, i.e 1/2

Thus demonstrating that if sin x = sin y it is not necessarily true that x = y.

The value sine function repeats every 360 degrees or 2 Pi.

That means; pick an angle, any angle, and take the sine. Instead add 360 degrees to the angle and take the sine again --> same answer.
ie: sin x = sin (x + 360)

If we have sin 2x, we only have to add 180 to x to achieve the same result.

sin 2x compared to sin 2(x+180) = sin (2x+360), so the function will repeat its value every 180 degrees.

If you have a function like y = sin x + sin 2x, one half repeats every 360 degrees while the other half repeats every 180 degrees. The combined function will repeat every 360 degrees - the longer of the two.

One part of one of the options does not repeat so regularly.



It is when we raise

 

[Saitama]

You can see that if you look at the graph of the sine function.

The usual way to solve these equations, however, is by looking at the unit circle.
See http://en.wikipedia.org/wiki/Unit_circle" .

On the unit circle you can see that there are 2 angles, where sine takes on the same value.
That is, for a small angle t, the sine of phi is the y-value on the right side of the unit circle belonging to this angle t.
But on the left side of the unit circle there is a second point with the same y-value.
The corresponding angle is pi minus the angle phi.

Beyond that, adding a full period (2pi) of the unit circle, will also yield the same y-values again.

Oh got it! Thanks for the explanation!

Thus demonstrating that if sin x = sin y it is not necessarily true that x = y.

The value sine function repeats every 360 degrees or 2 Pi.

That means; pick an angle, any angle, and take the sine. Instead add 360 degrees to the angle and take the sine again --> same answer.
ie: sin x = sin (x + 360)

If we have sin 2x, we only have to add 180 to x to achieve the same result.

sin 2x compared to sin 2(x+180) = sin (2x+360), so the function will repeat its value every 180 degrees.

If you have a function like y = sin x + sin 2x, one half repeats every 360 degrees while the other half repeats every 180 degrees. The combined function will repeat every 360 degrees - the longer of the two.

One part of one of the options does not repeat so regularly.



It is when we raise

So if it is |sin2x|, the period becomes pi/2. Right...?
And what about if it is sin$\sqrt{x}$?

 

[PeterO]

Oh got it! Thanks for the explanation!
So if it is |sin2x|, the period becomes pi/2. Right...?

That is correct

What about if it is sin$\sqrt{x}$?

For this example, we really should use radians. The sine function repeats every 2.pi - let's approximate that to 6.

The sine 0 will have the same value as sin 2.pi and sin 4.pi and sin6.pi etc

With our approximation that means sin 0 = sin 6 = sin 12 = sin 18

Since we are using sin$\sqrt{x}$

What value of x gives 0? [that one is pretty easy]

What value of x gives 6?

What value of x gives 12?

What value of x gives 18?

Are those values of x evenly spaced?

 

[Saitama]

For this example, we really should use radians. The sine function repeats every 2.pi - let's approximate that to 6.

The sine 0 will have the same value as sin 2.pi and sin 4.pi and sin6.pi etc

With our approximation that means sin 0 = sin 6 = sin 12 = sin 18

Since we are using sin$\sqrt{x}$

What value of x gives 0? [that one is pretty easy]

What value of x gives 6?

What value of x gives 12?

What value of x gives 18?

Are those values of x evenly spaced?

For 0, x=0
For 6, x=36
For 12, x=144
For 18, x=324

No the values of x are not evenly spaced. So that means sin$\sqrt{x}$ is not a periodic function. Right..?

 

[PeterO]

For 0, x=0
For 6, x=36
For 12, x=144
For 18, x=324

No the values of x are not evenly spaced. So that means sin$\sqrt{x}$ is not a periodic function. Right..?

Exactly.
Notice how you can get a feel for the funtion without using exact values. If the spacings between the numbers had been "approximately" even, you would have to go through the "pain" of using exactly 2.Pi to be sure.

 

[Saitama]

Exactly.
Notice how you can get a feel for the funtion without using exact values. If the spacings between the numbers had been "approximately" even, you would have to go through the "pain" of using exactly 2.Pi to be sure.

Lets get back to the question, the (a) option is |sin(3x)|+sin²x.
The period of |sin(3x)| is pi/3 and the period of sin²x is pi.
Is this function periodic?

 

[PeterO]

Lets get back to the question, the (a) option is |sin(3x)|+sin²x.
The period of |sin(3x)| is pi/3 and the period of sin²x is pi.
Is this function periodic?

Yes it is. The Period is pi. in the 0 ti pi range, the first function goes through 3 cycles, while the second goes through 1.

If a function consists of the sum or difference of two periodic functions, it will itself be periodic.

 

[Saitama]

Yes it is. The Period is pi. in the 0 ti pi range, the first function goes through 3 cycles, while the second goes through 1.

If a function consists of the sum or difference of two periodic functions, it will itself be periodic.

Are there some tricks to remember whether a given function is periodic or not?
Like you said "If a function consists of the sum or difference of two periodic functions, it will itself be periodic." If it was like this, one of the function is periodic and other is not, then what would be the result?

 

[PeterO]

Are there some tricks to remember whether a given function is periodic or not?
Like you said "If a function consists of the sum or difference of two periodic functions, it will itself be periodic." If it was like this, one of the function is periodic and other is not, then what would be the result?

As soon as one is not periodic, the whole thing isn't.

The easiest ones to recognise as non-periodic are those where the argument is an indice.

In sin (x) , sin is the function, x is the argument

sin (x^2) is not periodic [remember x^2 is an indice]

What ever you do, don't confuse (sin x ) ^2 and sin (x^2)

If you used a calculator to evaluate those two , then for the first you would take the sign of the angle [getting an answer between -1 and 1] then square the answer. For the second you would first square x, then plug that answer into the sine function.

 

[Saitama]

As soon as one is not periodic, the whole thing isn't.

The easiest ones to recognise as non-periodic are those where the argument is an indice.

In sin (x) , sin is the function, x is the argument

sin (x^2) is not periodic [remember x^2 is an indice]

What ever you do, don't confuse (sin x ) ^2 and sin (x^2)

If you used a calculator to evaluate those two , then for the first you would take the sign of the angle [getting an answer between -1 and 1] then square the answer. For the second you would first square x, then plug that answer into the sine function.

What is an indice?

 

[PeterO]

What is an indice?

There is a whole chapter on them, including the laws to use with indices, in every maths book.
The laws are sometimes called index laws.
An indice has a base and an index.
Common indices can have x as the base and a number as the index, often written here as x^2 or x^3