Is $180^\circ = \pi$?

Question

I want to ask a question that causes confusion. In the trigonometry, we use some units of measure of angle: degree and radian. Which is/are correct?

$$ 180^{\circ} = \pi $$

or

$$ 180^{\circ} = \pi \ \ \ \text{radians}$$

In the other words: Is $180^\circ$ a real number?

My opinion $180^\circ$ is real number like $3.14$. Thanks for your comments/answers.

Edit: Some my friends high school mathematics teachers present in the form of a real number of $180^\circ$ in the class, my other friends comment: "No, this is not a number. Because they have units". I said that '' We can calculate $\sin(x)$ function at the point $x = \pi/4$. Hence this argument tells us that $x$ is a real number. Because, the sine function is defined in real numbers''. Also, there are a third group. They says ''Don't worry about these! It doesn't matter what this concept is, we can make a calculation and go on our way''.

We could not agree on how to present this concept to students correctly.

Answer 1

It is possible to treat degrees and radians as units, just as we treat inches and centimeters as units. Just as we can convert inches to centimeters (1 in = 2.54 cm), we can convert degrees to radians (1 degree = $\pi/180$ radians).

But this point of view overlooks an important fact about radians: The radian measure of an angle is the ratio of two lengths (arc length divided by radius), so as a "unit" it is a pure number (cm/cm, or in/in, or furlong/furlong) independent of any arbitrary choice of units for measuring lengths (or masses or anything else). Because of this, I would prefer to identify "radian" with the pure number $1$. As a consequence, I would identify "degree" with $\pi/180$. And I would therefore answer the title question affirmatively: $180^\circ=\pi$.

Answer 2

Both $180$ and $\pi/4$ are real numbers.

Units are an extramathematical notion related to measurement and indicating how to interpret a real number in some given (for example physical) context.

One can speak of $\pi/4$ degrees and $180$ radians just as one speaks of $\pi/4$ radians and $180$ degrees.

One definition of the sine function is that it is the unique solution of the initial value problem, $\tfrac{d^{2}x}{dt^{2}}= -x$, $x(0) = 0$, $\tfrac{dx}{dt}(0) = 1$ (cosine is defined similarly, with different initial conditions). What might be called the natural units of this solution, $x(t) = \sin{t}$, are radians and not degrees. The functions $s(t) = \sin(\tfrac{2\pi}{360}t)$ and $c(t) = \cos(\tfrac{2\pi}{360}t)$ are what what might be called the "sine in degrees" and "cosine in degrees". However, note that their derivatives no longer satisfy the usual identities. For example, $s^{\prime}(t)$ is not the cosine in degrees function, rather it is a multiple of it, namely $s^{\prime}(t) = \tfrac{2\pi}{360}\cos(\tfrac{2\pi}{360}t) = \tfrac{2\pi}{360}c(t)$. That is $c^{\prime}(t) \neq -s(t)$ and $s^{\prime}(t) \neq c(t)$. The function $s(t)$ solves the initial value problem $\tfrac{d^{2}x}{dt^{2}}= -\left(\tfrac{2\pi}{360}\right)^{2}x$, $x(0) = 0$, $\tfrac{dx}{dt}(0) = \tfrac{2\pi}{360}$.

Put another way, although $(\sin^{\prime}(t), \sin(t))$ parameterizes the unit circle, it is not true that $(s^{\prime}(t), s(t))$ parameterizes the unit circle. If we work in degrees rather than radian identities such as the Pythagorean identity $\sin^{2}(t) + \cos^{2}(t) =1$ fail in the sense that it is not true that $s^{2}(t) + (s^{\prime}(t))^{2}$ equals $1$.

Alternatively, the functions $(\cos(t), \sin(t))$ in radians yield a unit speed parameterization of the unit circle, whereas the functions $(c(t), s(t))$ in degrees only yield a constant speed parameterization of the unit circle. The choice of units corresponds to the choice of normalization of the speed of the parameterization. The non-unit speed parameterization is less aesthetically pleasing because it breaks the symmetry of certain standard identities, and practically a bit unwieldy because of the factors of $\tfrac{2\pi}{360}$ it generates.

Answer 3

Radians are "pure numbers", inasmuch as they are ratios of the length along the arc subtending the angle to the radius of the circle. As such, they don't require units, and you could (and should) write $180^{\circ} = \pi$.

That being said, there's a good reason to specifically write radians. As Dan Fox points out that there is another natural unit for angle measure: the turn. In fact, the turn is arguably the most natural measure of an angle, since everyone understands what it is (though our language confuses the issue: "Turn around" and "Turn halfway around" are used interchangeably).

This is also a ratio (it's the fraction of a full rotation) and as such has no units. So you could just as legitimately write $180^{\circ} = 1$.

Unfortunately, this leads to writing $1 = \pi$.

So while you could write $180^{\circ} = \pi$ and not raise an eyebrow, it's probably better to write $180^{\circ} = \pi$ radians.

This is how I introduce angles to my trigonometry students.

Answer 4

$180^{\circ} = \pi$?

Certainly not : the surface of the unity circle is $\pi$. If anyone tells me the surface of anything can be expressed in degrees, I'll tell this person (s)he's wrong.

$180^{\circ} = \pi$ radians?

That I can agree to.