# Why do trigonometry angles go counter-clockwise? (1 Viewer)

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#### Banaticus

The idea of a circle divided into 360 degrees that goes clockwise has been around for thousands of years (see orienteering compasses). So, why do radians and angles in trigonometry go counter-clockwise and start off pointing to the right? Was that on purpose?

#### HallsofIvy

That is solely a convention. It is, fundamentally, just because someone decided to make clocks in which the hands move, well, "clockwise"!

#### AlephZero

Homework Helper
It is, fundamentally, just because someone decided to make clocks in which the hands move, well, "clockwise"!
In the northern hemisphere, that convention is "built in" to sundials because of the happenstance of the solar system, so I guess the first clockmakers just stuck with it.

On the other hand, the sign convention for angles is linked to using a right handed coordinate system for 3-D geometry. Again that is just a convention, but there are very few exceptions to it in math physics or engineering.

#### Banaticus

Why would a clockwise (with 0 at the top, as in compass degrees) or a counterclockwise (with 0 on the right as in trigonometry degrees) coordinate system be considered left or right handed?

#### AlephZero

Homework Helper
A rotation from the X axis to the Y axis is taken as rotation in the positive direction around the Z axis.

#### DaveC426913

Gold Member
Agree with AlephZero. The origin is based on the cartesian xy plane. X values go from 0,0 positively to the right, and a line on an XY grid will extend from 0,0 to the right. Positive angles from this line will move into the +X, +Y quadrant and so will rotate about 0,0 counterclockwise.

#### Banaticus

Yes, but where did this convention come from? Even today, orienteering follows the practice that clockwise around a Z-axis is positive. Why did it change? Maybe some pictures will help me explain what I'm asking. For thousands of years, this was the standard convention -- positive angles went clockwise and started from the top (the Y-axis, although it wasn't called the Y-axis then), like this picture:

But trigonometry doesn't follow that convention, it reverses it and has positive angles go the opposite way (counter-clockwise) from the X-axis, like this picture:

(Edit: Drawings are not to scale.)

Trig quadrants today have angles that start on the X-axis and go counter-clockwise like this:

Why don't they start on the Y-axis and go clockwise like this:

People were starting from the "top", from the Y-axis, for thousands of years, why did it switch, why do trigonometry angles go counter-clockwise? Why aren't positive angles in trigonometry measured clockwise around a Z-axis, as they still are in orienteering today? Who created this trigonometry quadrant stuff and why'd they make it different?

#### gsal

Banaticus: Don't get too hang up on these things!

Clearly, you are a fan of orienteering; but that does not mean that trigonometry switched the convention.

First of all, it looks like orienteering is not even 200 hundred years old...sure, they use compasses and the compass is 2000 years old...but so is trigonometry. By the way, it looks like the compass was invented in China and introduced to Europe 150 years later...which means that trigonometry had a head start in Europe.

By the way, I had never heard of orienteering until this post...o.k., so I don't get out too much...but I have been hearing about trigonometry all my life! So, why orienteering changed the convention?!! .. just kidding...no need to answer.

In any case, the use of each convention is probably to facilitate its use within the corresponding application...let's get practical, here.

my 2 cents.

#### LegendLength

I think the question your really asking is why does the X axis go to the right and the Y axis upwards. Because if the Y axis pointed downwards (for positive direction) then you'd have the compass style of angles you talk about.

When people draw graphs they often only need the positive parts of the axis because real world objects are rarely negative. And I would say the most natural way to do that is to have a line down the left and a line along the bottom of the graph. The reason I would give for that is it allows bar graphs to 'sit up' from the bottom rather than 'hang down' from the top, as you would have if the Y axis went downwards instead of upwards.

#### Deveno

i'm not sure of the answer. one possibility is that much of early trigonometry derives from indian, babylonian and arabic sources, so a counter-clockwise choice for "positive" angle orientation may have seemed more natural to them.

it is unlikely that cartesian coordinate systems are to blame, as rene descartes did not come along until the 17th century, and by then trigonometric conventions would have been in place for several centuries.

in any case, a choice of orientation is an arbitrary one, and if you prefer, you are free to adopt one of your choosing (although you may wish to indicate this choice in any written or spoken communication that is based on your personal convention).

the "right-hand rule" is clearly an attempt of the right-handed majority to impose their will upon the left-handed minority, and may very well be a violation of civil rights, in countries that have such legislation.

#### dalcde

Maybe its how people normally draw right triangles that lead to that "trigonometric" convention.

#### jedishrfu

Mentor
It's not just orienteering it's also a naval convention

Wikipedia has some discussion under clockwise that might shed light on these conventions.

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