# Why do trigonometric graphs with a π don't take inputs in radians/degrees?

• autodidude
In summary, the reason for using radians instead of degrees in trigonometric functions is due to the convenience and consistency it provides in various mathematical applications, such as engineering and calculus. Radians are also more closely related to the concept of angles and unit circles, making them a more natural choice for measuring and calculating trigonometric values. Additionally, using radians allows for easier manipulation and simplification of trigonometric identities and equations.
autodidude
e.g. I want to find y when x is at pi/2 for the graph y=cos pi x. Why does pi/2 have to be expressed as a 'normal' (for want of better word) number (3.14/2) and not as 90 degrees (180/2) like you would for a graph without a pi in it

e.g. cos x
For cos x, if x is at pi/2, then I just put in pi/2 in degrees

I get the feeling that the explanation is super obvious but I just don't see it yet

hi autodidude!

(have a pi: π and a degree: ° )

there's an engineering reason and a calculus reason for using radians (π etc) instead of degree …

(i expect other people will jump in with other reasons too! )

i] if θ is in radians, then rθ is the distance a wheel travels, and rdθ/dt and rd2θ/dt2 are the speed and acceleration of the wheel (or of a point at distance r from the centre of any rotating body)

ii] cosx is the real part of eix, and d(cosx)/dx = -sinx

(but you can write cos180° if you want to … i certainly think "180°" whenever i see "π" … and nobody ever writes "cos1.57" when they mean "cosπ/2"! )

It depends on how you define the trigonometric numbers. You can define them either by using angles or power series(or the euler formula).

If you use angles, you can also use the circumference of a circle and define radians. However, that way sinx will be defined as long as x is either degrees or angles

If you use power series, you can prove all trigonometric identities without even defining angles.
However,that way sinx will be defined as long as x is a number

^ Thanks guys but I'm not sure that answers my question...or I just don't understand what you're saying

What I mean is if it's y=cos x, to find y, I'd just pick a point on the graph whose x-coordinate would be represented as π/4, π/2, π etc. Now when I plug it into x, I express it in °, so if x=π/4, then I'd put in cos(45°) and that would give me the point for y

But if the graph is y=cos πx and I want to find y when x is at π/4, I have to treat the input as 3.14/4 rather than 180°/4

I'm not really following you, but if you want to use pi/2 etc. then make sure your calculator mode is on radians.

Basically I just want know why x-inputs cannot be in degrees when there is a π in the equation

---

y=cos πx

When x=π/4, if you use degrees then you get 180/4, so plug that in

y = cos(180 x (180/4))
y = cos(8100)
y = -1

But you have to use 3.14/4 and not 180/4 for it to be consistent with the graph

y = cos(180 x (3.14/4))
y = cos(141.3)
y = -0.780

I think you are a bit confused. If the equation is $$y=cos\pi x$$ then (I think) π is an angle, in radians. If so, x is just a number (not an angle) because it doesn't make much sense to multiply an angle by an angle. If pi is just a constant, then turning it into 180 doesn't make much sense as well. Anyway, if you evaluate
$$\cos(\frac{180\pi}{4})$$
since you are working with radians, make sure you get your calculator in radian mode! In radian mode, it does evaluate to -1.

By the way, can you provide the context (if any) since it does make a difference whether π is an angle and x is a number, or π is a number and x is a constant.

hi autodidude!

(just got up :zzz: …)
autodidude said:
Basically I just want know why x-inputs cannot be in degrees when there is a π in the equation

---

y=cos πx

When x=π/4 …

ah, i think you're misunderstanding how radians work …

y = cosx does have x in radians

y = cosπx doesn't … the x there is in "half-revolutions", and is sometimes more convenient

so you wouldn't have y = cosπx and x = π/4, you'd have y = cosπx and x = 1/4

I'll toss my 2 cents worth into the fray. You are really talking about two different functions.

f(x) = sin(x) with x in radians
g(x) = Sin(x) with x in degrees (capital letter to distinguish them).

They both accept any number as their argument, and the relation between them is that

S(x) = sin(t) where t = Pi*x/180.

And you can put pi in either one. s(pi) is the same value as S(pi*180/pi) = S(90) and S(pi) is the same value as s(pi2/180) = s(.05483). You just never see the second one because pi isn't a convenient number to use when expressing in degrees.

And, of course, sin(x) is the one whose derivative is cos(x), whereas the derivative of Sin(x) is not Cos(x).

LCKurtz said:
s(pi) is the same value as S(pi*180/pi) = S(90)

That should be S(180) not S(90). Dunno why it won't let me edit it.

dalcde said:
I think you are a bit confused. If the equation is $$y=cos\pi x$$ then (I think) π is an angle, in radians. If so, x is just a number (not an angle) because it doesn't make much sense to multiply an angle by an angle. If pi is just a constant, then turning it into 180 doesn't make much sense as well. Anyway, if you evaluate
$$\cos(\frac{180\pi}{4})$$
since you are working with radians, make sure you get your calculator in radian mode! In radian mode, it does evaluate to -1.

By the way, can you provide the context (if any) since it does make a difference whether π is an angle and x is a number, or π is a number and x is a constant.

So, if π in y=cosπx is in radians and x is just a number, say 2, then you'd have y=cos3.14(2)..? Or am I misunderstanding you and you mean it's in radians, as in represented in π radians but when you evaluate it you use degrees?

That's sort of what I was getting at, multiplying an angle by an angle! Part of the answer I was looking for was why it doesn't make sense haha. I'm unfamiliar with the different modes on my calculator..in radian modes, it treats π as 180 right?

I was just asked to state the period and amplitude of y=(cos(πx/2)/4 and I tried graphing it out of curiosity. I was plugging in radians for x (and treating the π in the equation as 180 degrees) and I kept getting only y=0, y=-0.25 and y=0.25 so I played around with some simpler equations and opened this can of worms

tiny-tim said:
hi autodidude!

(just got up :zzz: …)

ah, i think you're misunderstanding how radians work …

y = cosx does have x in radians

y = cosπx doesn't … the x there is in "half-revolutions", and is sometimes more convenient

so you wouldn't have y = cosπx and x = π/4, you'd have y = cosπx and x = 1/4

y=cosx has x in radians...could you please elaborate? Do you mean that you can use radians as inputs but the calculator has to be in radian mode?

I THINK I understand what you mean for that second part...am I correct in thinking of it this way, because the constant is half a revolution (or angle?), then x is the number of half revolutions (or 180°s..?) which is why x shouldn't be another angle?

LCKurtz said:
I'll toss my 2 cents worth into the fray. You are really talking about two different functions.

f(x) = sin(x) with x in radians
g(x) = Sin(x) with x in degrees (capital letter to distinguish them).

They both accept any number as their argument, and the relation between them is that

S(x) = sin(t) where t = Pi*x/180.

And you can put pi in either one. s(pi) is the same value as S(pi*180/pi) = S(90) and S(pi) is the same value as s(pi2/180) = s(.05483). You just never see the second one because pi isn't a convenient number to use when expressing in degrees.

And, of course, sin(x) is the one whose derivative is cos(x), whereas the derivative of Sin(x) is not Cos(x).

S(pi) = S(180°)?

's(pi^2/180) = s(.05483)', input of radians...?

I think part of the reason why I'm getting confused here is because I only use my calculator in ° mode...

hi autodidude!
autodidude said:

i mean that cos45 means "cos of 45 radians" …

if you want degrees, you need to write them in, eg cos45°
… for that second part...am I correct in thinking of it this way, because the constant is half a revolution (or angle?), then x is the number of half revolutions (or 180°s..?) which is why x shouldn't be another angle?

exactly!
I think part of the reason why I'm getting confused here is because I only use my calculator in ° mode...

that's fine, most physics exam questions do use degrees rather than radians

(oooh, except rolling wheel questions, where you need radians so that you can write v = rω etc )

autodidude said:
S(pi) = S(180°)?

No, that should be s(pi) = S(180) (lower case s)

's(pi^2/180) = s(.05483)', input of radians...?
Yes.
I think part of the reason why I'm getting confused here is because I only use my calculator in ° mode...

When your calculator is in degree mode you are calculating S(x) and when you have it in radian mode you are calculating s(x).

Ah I got you both...many thanks! Now I can move on to the next questions...

## Related to Why do trigonometric graphs with a π don't take inputs in radians/degrees?

The use of radians in trigonometric graphs is due to the fact that radians are a unit of measurement that directly relates to the geometry of circles. This makes them a more natural choice for representing angles in trigonometry, as they allow for simpler calculations and relationships between angles and the unit circle.

## 2. Can I use degrees instead of radians in trigonometric graphs?

Yes, you can convert degrees to radians by using the conversion factor of 180 degrees = π radians. However, it is recommended to use radians in trigonometric graphs as they provide a more intuitive representation of angles and are the standard unit of measurement in mathematics.

## 3. Why do some trigonometric functions use degrees and others use radians?

This is because different trigonometric functions have different domains and ranges, which determine the unit of measurement used for their inputs and outputs. For example, the sine and cosine functions have a domain and range of angles, so they use radians as the unit of measurement. However, the inverse trigonometric functions have a domain and range of ratios, so they use degrees as the unit of measurement.

## 4. Do radians and degrees give the same results in trigonometric graphs?

No, radians and degrees do not give the same results in trigonometric graphs. This is because they represent angles in different ways, with radians being a unit of measurement on the unit circle and degrees being a unit of measurement on a 360-degree circle. Therefore, the same angle in radians and degrees will have different numerical values.

## 5. Why is π used in trigonometric graphs instead of other values?

The value of π is used in trigonometric graphs because it is a fundamental constant in mathematics and has a special relationship with circles. It represents the ratio of a circle's circumference to its diameter and is a key component in many trigonometric formulas and identities.

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