# How to "think in radians"

Nonetheless, how do you recommend I do this? It's hard to have an alternative to degrees since I already feel so comfortable working in degrees.

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berkeman
Mentor
That's a good question. I usually think in terms of the unit circle, but that's because I deal with complex numbers and complex functions so much in my EE work. Have you learned about complex numbers and the unit circle yet? Where the x-axis is the Real axis, and the y-axis is the Imaginary axis? As you work with complex exponentials and their equivalent time-dependent trig functions, you work pretty much exclusively with radians as the arguments to those functions.

SteamKing
Staff Emeritus
Homework Helper
It sounds like you didn't fully grasp the definition of the radian.

1 radian is the angle which is subtended by an arc which has the same length as the radius of the circle. Since the ratio of the circumference of a circle to its diameter is equal to pi, by definition, then the radian measure of one circuit of the circle will be 2pi radians.

Since there are pi radians in a semi-circular arc, which is also 180 degrees, then the radian measure of an angle expressed in degrees = pi * angle / 180.

IDK where your teacher is from, but I'm an engineer, and most technical people with whom I've worked use degrees and convert to radians when the need arises. IDK anyone who asks to use a 'pi/4 - pi/4 - pi/2' triangle (45-45-90) or who goes to the next stop sign and makes a pi/2 turn. Radian measure is primarily one of convenience, as it simplifies certain math concepts, particularly for trig functions, in calculus and other types of math study. Compass bearings and navigational coordinates (latitude and longitude) still use degrees, or degrees, minutes, and seconds. Even astronomers haven't abandoned degrees, minutes, and seconds.

You could ask your teacher if he/she uses a conventional watch (with hours, minutes, and seconds) or some special hipster decimal time piece.

Ray Vickson
Homework Helper
Dearly Missed

Nonetheless, how do you recommend I do this? It's hard to have an alternative to degrees since I already feel so comfortable working in degrees.
Probably you feel comfortable using degrees because you have used them a lot. So, to get comfortable with radians, just use them a lot also; that is, get a lot of practice using them. Believe it or not, after a while they (radians) will not seem so strange after all.

That's a good question. I usually think in terms of the unit circle, but that's because I deal with complex numbers and complex functions so much in my EE work. Have you learned about complex numbers and the unit circle yet? Where the x-axis is the Real axis, and the y-axis is the Imaginary axis? As you work with complex exponentials and their equivalent time-dependent trig functions, you work pretty much exclusively with radians as the arguments to those functions.
We haven't done any of that yet. I have seen instances where Y is used as an imaginary axis and X as a real but I didn't really looked into it.

It sounds like you didn't fully grasp the definition of the radian.

1 radian is the angle which is subtended by an arc which has the same length as the radius of the circle. Since the ratio of the circumference of a circle to its diameter is equal to pi, by definition, then the radian measure of one circuit of the circle will be 2pi radians.

Since there are pi radians in a semi-circular arc, which is also 180 degrees, then the radian measure of an angle expressed in degrees = pi * angle / 180.

IDK where your teacher is from, but I'm an engineer, and most technical people with whom I've worked use degrees and convert to radians when the need arises. IDK anyone who asks to use a 'pi/4 - pi/4 - pi/2' triangle (45-45-90) or who goes to the next stop sign and makes a pi/2 turn. Radian measure is primarily one of convenience, as it simplifies certain math concepts, particularly for trig functions, in calculus and other types of math study. Compass bearings and navigational coordinates (latitude and longitude) still use degrees, or degrees, minutes, and seconds. Even astronomers haven't abandoned degrees, minutes, and seconds.

You could ask your teacher if he/she uses a conventional watch (with hours, minutes, and seconds) or some special hipster decimal time piece.
I do remember seeing a clock in a math classroom last year that wasn't told by time, but radians of the circle. Had no idea what it meant until we actually learned about radians.

I think I just have to get myself in the habit of using them. I want to be able to picture what 5pi/3 radians or 12pi/7 radians would look like without having to convert to degrees.

vela
Staff Emeritus
Homework Helper
I wouldn't necessarily call converting to degrees every time a bad idea. It's just a waste of time.

Your question is like asking someone how to think in centimeters because you're used to working in inches. There's really no thinking involved. It's just a different unit. The more you use it, the more familiar it becomes.

HallsofIvy
Homework Helper
Memorize a few basic "landmarks". A full circle is $2\pi$ radians. Half a circle is $\pi$ radians. A quarter circle (a right angle) is $\pi/2$ radians. It is also worth memorizing that the two equal angles in an isosceles right triangle are $\pi/4$ radians, that the three congruent angles in an equilateral triangle are $\pi/3$ radians, and that the acute angles in the two right triangles you get by dividing and equilateral triangle into two parts are $\pi/3$ and $\pi/6$

That is:
360 degrees= $2\pi$ radians
180 degrees= $\pi$ radians
90 degrees= $\pi/2$ radians
60 degrees= $\pi/3$ radians
45 degrees= $\pi/4$ radians
30 degrees= $\pi/6$ radians

For other angles, use x degrees= $(x/180)\pi$ radians.

berkeman
Mentor
There is some mathematical operation that only works in radians -- does anybody remember what that is?

HallsofIvy
Homework Helper
The derivative formulas for the trig functions are derived assuming that the variable represents "radian" measure. If the variable is in degrees, the formulas would be multiplied by non-unit constants.

(Actually, there is a little more to it than that. Strictly speaking the "x" in the functions sin(x), cos(x), etc. is NOT an angle at all. It is simply a "variable", a number, and has NO units. In order to match that variable to angle values that engineers use, in tables or calculators, we have to treat it as if it were in radians.)

marcusl
Gold Member
One good reason not to convert to degrees is so you can use small-angle approximations for trigonometric functions. For example, $sin(\theta)\approx\theta$ for $\theta<<1$, holds only when theta is measured in radians.

Nonetheless, how do you recommend I do this? It's hard to have an alternative to degrees since I already feel so comfortable working in degrees.
Sometimes the best you can do is bite the wax tadpole and memorize the blasted stuff. Likewise, I prefer to cheat and change everything to degrees (incredibly simple with any modern calculator).
But....for the time being, obey your professor. It probably won't kill you. :)

olivermsun
I think I just have to get myself in the habit of using them. I want to be able to picture what 5pi/3 radians or 12pi/7 radians would look like without having to convert to degrees.
Well, the "landmarks" that HallsofIvy posts above are useful to have in your head. It's like knowing that 30, 60, 90°, …, 270°, 360° occur at different parts of the circle.

Other than that, since ##\pi## radians is a semicircle, then you can think of any fraction in terms of parts of the semicircle, e.g., ##5\pi/3## is just the semicircle plus ##2/3## of the other semicircle.

NascentOxygen
Staff Emeritus
##360^o = 2\pi^c##

Simon Bridge
Homework Helper
radians are usually taught by definition ... like rad=deg times pi divided by 180.
Is that how your class did it?

In fact the radian comes naturally from exploring the concept of an angle.
I usually let my students define their own way to measure angles, and discuss it.
They usually end up with radians ... then wonder why their protractors are marked out in those annoying degrees.

Here's the idea in a nutshell:

Every measure has a special unit shape to deal with it.
For instance, distance uses a unit length - we pick a line segment and say it's length is "1 unit"; the length of a line is the number of those line segments that fit in it. We may choose 1cm as the unit or 100cm or 5cm or a foot or a mile or a number of wavelengths of some sort of radiation. Get the idea?

Area uses a unit square and volume uses a unit cube in the same way.
We define these unit shapes in terms of what we decided for the unit line segment in order to keep things consistent, so the unit square has sides equal to the unit length. We could have decided that the unit square would have a diagonal distance equal to the unit length but that's not so convenient.

The natural shape to use to define an angle is the circle ... so you need a unit circle.
A unit circle can be defined by radius = 1, diameter = 1, or circumference =1. Which one is most convenient?

The size of an angle is then defined to be the length around the circumference that sits inside the sides of the angle.

It is easiest to draw a circle by setting the radius - you do it all the time with a compass.
eg. if we pick the length unit as 5cm, then the unit circle has a radius of 5cm.
We measure the size of an angle by putting the center of the unit circle on the corner, and measuring around the circumference ... if the distance between the sides is, say, 3cm, then the size of the angle is 3/5.
There is a drawback though - if the angle is a right-angle, say, then the distance around the circumference is 1/4 the total circumference ... and the total circumference of a circle with radius = 1 is...
There are advantages though - for instance, it means that the arclength inside an angle of a circle radius R is ##s=R\theta##

Making a circle with unit circumference is tricky - it can be done i.e. by getting a unit length strip of metal and bending it round into a circle.

Using the circumference = 1unit version means that all the angles will be less than 1, a right angle would be 1/4 ... people don't like working in fractions, so we divide the circumference into sub-units and choose the number of sub-units in a circle to be a big number that divides easily by a lot of other numbers so pretty much every angle we want to measure will have a nice number bigger than 1. A useful figure that has stood the test of time is 360. So a right angle would be 90. Sound familiar?
If we use this, the arclength for a circle radius r is given by ##s=\pi r \theta /180##.
The other one is simpler in some ways while this one is simpler in others.

As you advance, you will use radians more than degrees - eventually you'll only use degrees to talk to people like builders.
You certainly should get used to using radians by default - whenever you can.
The easiest conversion is to think what proportion of a circle fits into the angle, then multiply that by 2 and write a pi after it.
So if 1/6th of the circle fits into the angle, then the angle is (2/6)pi radians.

Doug Huffman
Gold Member
There is more to the unit circle and more to be taken from memorizing it than merely cribbing degrees to radians. The trigonometric functions are nicely regular at the cardinal and inter-cardinal points.

I am traveling and away from my 45 y.o. class notes. When I am home and if I remember, I'll try to sketch the construction of the UC that I memorized. No. Here it is ...