Why can't degrees be used to approximate tan(46)?

[Prem1998]

I was approximating tan46 using derivatives. If I do it using radians, then we know the value of the function at pi/4, and the difference, i.e. dx is 1 degree=0.01745 radians.
It's derivative at x=pi/4 is 2.

So, approximate change in the value of the function is= 2*0.01745
. =0.0349
So, tan46=y+dy= 1+0.0349=1.0349, which is close to the actual value

BUT, this happens when I try to use degrees:
The derivative at x=45 remains the same but the difference is 1 degrees
So, dy=2*1=2
So, tan46=y+dy= 1+2 =3

Why am I getting a wrong answer just by changing the units? Degrees and radians are just multiples of each other, right? Units are just relative. 1 Newton is not 'better' than 1 dyne, right? The calculation of trigonometric derivatives from first principles doesn't assume that x should be in radians. Any step which I have done in the approximation is not radian dependent. Then, why're degrees giving wrong approximation?

 

[Doc Al]

The calculation of trigonometric derivatives from first principles doesn't assume that x should be in radians.

Actually, it does. At least if you want the usual results, e.g., that the derivative of sin(x) = cos(x).

 

[Prem1998]

Actually, it does. At least if you want the usual results, e.g., that the derivative of sin(x) = cos(x).

I don't think so. Can you tell that which step in the calculation of derivative of sin(x) can't be done if x is in degrees? Both are just units. Degrees give complete revolution a value of 360 and radians give 2pi, then what's the difference?
Are you saying that the graph of sin(x) doesn't have slope equal to cos(x) if degrees are used as unit for the x-axis?

 

[Doc Al]

Can you tell that which step in the calculation of derivative of sin(x) can't be done if x is in degrees?

When deriving the derivative of sinθ there is a step that uses the limit of sinθ/θ as θ goes to zero. That limit = 1 only if θ is measured in radians.

Are you saying that the graph of sin(x) doesn't have slope equal to cos(x) if degrees are used as unit for the x-axis?

That's right.

 

[willem2]

Functions in mathematics do not have units. sin(x) with x in radians is just a different function from sin(x) with x in degrees. in partiular sin_degrees(x) = sin_radians (π/180 x).
unsurprisingly, the different functions have different derivatives.

 

[Prem1998]

When deriving the derivative of sinθ there is a step that uses the limit of sinθ/θ as θ goes to zero. That limit = 1 only if θ is measured in radians.That's right.

You're right about the limit but how can the slope change?
Suppose we graph sin(x) on a graph paper using radians. Then, isn't switching to degrees just like only magnifying the graph? The point which we were calling 1, we now divide it into 57 parts and call each part 1 unit. Magnifying doesn't change the graph and therefore doesn't change the slopes.
In other words, if we draw a really big graph and mark the x- axis with radian values and then starting from 0 we write 1 degree at the point corresponding to 0.0174 and write graduations in the same way, then that won't change the graph, right?
As a side question, what would be the derivative of sinx if x is in degrees?

 

[Doc Al]

You're right about the limit but how can the slope change?

Realize that slope is dy/dx and thus has "units". If you change the units used for x, you get a different function and a different slope.

As a side question, what would be the derivative of sinx if x is in degrees?

Let's call that function sin_d(x), where x is in degrees. Express it in terms of the normal sine function in radians: sin_d(x) = sin(πx/180).

Now just use the chain rule. sin'_d(x) = sin'(πx/180) = (π/180)cos(πx/180) = (π/180)cos_d(x)

 

[Erland]

If you draw a graph of y = sin x using radians and another graph of y = sind x (where we use degrees), and we have the same scales on the axes in both cases, then you have two entirely different graphs, one with period 2π and one with period 360. Therefore, the slopes will also be entirely different.

 

[Doc Al]

If you draw a graph of y = sin x using radians and another graph of y = sind x (where we use degrees), and we have the same scales on the axes in both cases, then you have two entirely different graphs, one with period 2π and one with period 360. Therefore, the slopes will also be entirely different.

Exactly.

 

[Stephen Tashi]

You're right about the limit but how can the slope change?

Perhaps the simplest example is to compare the slope of the functions y = x and y = 2x.

For a physical example, suppose the force in Newtons exerted by a spring when it is compressed x centimeters is given by the equation y = x, which we should write as y = (1 Newton/centimeter) x to make the units on the left and right hand side of the equation "balance".

If we measure distance in meters, the same physical situation is described by the equation y = 100 x, which should be written as y = (100 Newtons/meter) x. The equations y = x and y = 100x have different slopes. But with the slopes (1 vs 100) there are associated physical units (1 Newton/cm vs 100 Newtons/m) that are equivalent.

You can think of sin(x) as describing a physical situation, but to create that physical situation, we need to construct an angle that has size "x". The relation of sin(x) to a physical situation is not completely described until you say how you will interpret x as a angle. The standard interpretation of sin(x) in a calculus book is that x will be constructed in radians.

To refer to the different physical situation where x is measured in degrees, we often abuse notation by using name "sin(x)" for that situation also, but really we ought to to name the function something else - or incorporate a dimensioned constant in it. (e.g. ##sin((1 \frac{\pi}{180}) x) ## )

 

[Prem1998]

If you draw a graph of y = sin x using radians and another graph of y = sind x (where we use degrees), and we have the same scales on the axes in both cases, then you have two entirely different graphs, one with period 2π and one with period 360. Therefore, the slopes will also be entirely different.

So, stretching a graph sideways causes the tangents at the points to gravitate towards x-axis, right? I visualize switching from 1 radian as one unit to 0.0174 radian as one unit, as stretching the same graph sideways.

 

[Prem1998]

Perhaps the simplest example is to compare the slope of the functions y = x and y = 2x.

For a physical example, suppose the force in Newtons exerted by a spring when it is compressed x centimeters is given by the equation y = x, which we should write as y = (1 Newton/centimeter) x to make the units on the left and right hand side of the equation "balance".

If we measure distance in meters, the same physical situation is described by the equation y = 100 x, which should be written as y = (100 Newtons/meter) x. The equations y = x and y = 100x have different slopes. But with the slopes (1 vs 100) there are associated physical units (1 Newton/cm vs 100 Newtons/m) that are equivalent.

You can think of sin(x) as describing a physical situation, but to create that physical situation, we need to construct an angle that has size "x". The relation of sin(x) to a physical situation is not completely described until you say how you will interpret x as a angle. The standard interpretation of sin(x) in a calculus book is that x will be constructed in radians.

To refer to the different physical situation where x is measured in degrees, we often abuse notation by using name "sin(x)" for that situation also, but really we ought to to name the function something else - or incorporate a dimensioned constant in it. (e.g. ##sin((1 \frac{\pi}{180}) x) ## )

Thanks, your example was really helpful.