Hey so i do 3 unit maths and i'm starting to get really confused when to use the radian mode and why do we exactly use it. I'm also starting to catch on that sometimes (mainly in ext), we use pi instead of 180 and the answer may drastically vary due to pi being 180 or 3.14.
So how do i know when to use pi as 180 or 3.14, AND when do i use radian mode and can i still complete a question without the radian mode.
For 3 unit the main thing you use degrees for are like 3D trig with like sine rule and cosine rule, finding the angle between two lines (but can still be done in radians unless specified).
Anything involving calculus like integrating, differentiating, use radians. Most questions use radians in 3 unit.
Also you can't "use pi as 180 or 3.14", pi radians is equal to 180 degrees it's not a changing value of pi. If you want 180 degrees and you're in radians then you just use pi instead.
Maybe post an example question and we can guide you through what to use.
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Sorry, nevermind what I said. I guess a geometrical issue arises when you try to graph a linear function on top of a trigonometric function that has been graphed using degrees as its input. This is because an input in degrees into a trigonometric function produces an output that is not in degrees. But an input in degrees into a linear function will not do this and since the y-axis isn't in degrees then it would be difficult to know where this output is located.
eg. an input of 90 degrees into f(x) = x outputs 90 degrees. But since the vertical axis is not in degrees then this output would actually be pi/2, hence why radians eliminates this geometrical issue.
Last edited by 1729; 4 Jul 2017 at 10:26 PM.
A "function" in the usual sense of the word assigns an element of the codomain (real numbers here) to every element of the domain (real numbers also), there is no reference to any particular units in the function or the plane itself. That's why lengths/areas/etc are measured in units/units^2/etc when we are doing coordinate geometry.
Calling the object that converts a degree measurement to a radian measurement "f(x)=x" is very misleading.
Deciding what we mean by "sin(x)" on the other hand is entirely a matter of definition, so we can rescale units as we wish. It is obviously pleasant that the derivative of a trig function is the complementary trig function up to sign, so we choose that scaling to do anything involving calculus.
Last edited by seanieg89; 5 Jul 2017 at 12:52 AM.
Yes I understand that no reference to units are made, hence why I said to nevermind what I said.
'Angle' is a ratio of the length of a circular arc to a radius. Both quantities involved in this ratio share the same dimensionality and so are cancelled out, naturally leaving 'angles' as dimensionless. It may be argued that degrees are not dimensionless but since the radian is dimensionless and the scaling factor between them is dimensionless, it follows that degrees too are dimensionless. Angles are dimensionless quantities but are still assigned units (eg. the Babylonian 'degree').
Being 'dimensionless' the angle measure is a 'pure number' but this number itself differs in magnitude between degrees and radians. Degrees can be considered to have lower magnitude because 360 are required for one revolution whilst only 2pi radians are required for one revolution.
The fact that there is no reference made to any particular units in a 'function' does not mean we cannot input an angle measure. Perhaps you mean that we cannot input a quantity with both dimension and units (eg. length 'three metres'). The input into a trigonometric function must be a pure number (a dimensionless quantity) and indeed angle measures are dimensionless. In fact, the value of the trigonometric function is also dimensionless.
And I didn't say f(x) = x converts degrees to radians. If we input 90 'degrees' how would we determine the output if we have chosen a different unit of angle measurement for the vertical axis (noting that these differing dimensional units have differing magnitude)? I said that if we input 90 'degrees' the output will also be 90 'degrees' but this does not fit onto the vertical axis.
Last edited by 1729; 5 Jul 2017 at 2:07 AM.
Although angles are dimensionless for the purposes of dimensional analysis, a choice of "angular unit" must be made in order for an angle to be considered as a real number, a necessary step in constructing trigonometric functions which we can use calculus on. (In fact when we do things rigorously it is quite a pain to properly axiomatise geometry, and the trig functions are usually defined by power series instead.)
Regarding your last paragraph in particular, what are we inputting 90 "degrees" into? The function f(x):=x defined on some domain D takes an element of the set D and spits out the same element. When we are doing calculus, we would like D to be a subset of real numbers, some Euclidean space or a manifold etc. This means that the choice of units for whatever physical/geometric object we are measuring occurs before the function gets introduced, so if we want to talk about the function that maps an angular measurement in degrees to an angular measurement in radians, this is usually done with the function f(x)=pi*x/180, not pi*x with some condition that the y-axis and x-axis have different scales.
All of this is mostly a matter of convention and mathematical practice, rather than a matter of any of our statements being "incorrect" so I don't really want to drag out this discussion further, but that's about all I have to say about it.
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