# MathHelp with a trigonometry function

#### fireballs619

##### Occam's Taser
Donator
Whilst doing a Geometry problem, I formed this function which inputs angle X and outputs angle Y. I currently have it written as follows;

$f(x)=\arcsin{\frac{\cos{2x}-\sin{2x}}{\sqrt{2}}}$

No code has to be inserted here.
The plot of this graph looks like this.

Having never been taught functions, I am a bit naive about whether or not this function is special, or what type of function it even is. I can tell from the table that as X increases, Y decreases until a point (where X is 22.5 degrees), at which point Y begins to increase along with X. I also know that, since it was derived from a geometry problem involving acute angles, X and Y cannot pass above 45 (since twice that is 90). Since, again, they are angles, X and Y are also limited to a lower limit of zero. Given this information, is anyone willing to help me find out more about this function? Any other ways to write it, what it tells me about the angles?

Pardon any ignorance in the above post, as I am working off of only what I have taught myself about functions, so some of it may not make sense.

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#### HarvesteR

##### Member
Hmm, what is it you want to know exactly?
It seems to be a normal linear function... is it supposed to be special for some reason?

I can tell you there is a very marked sine wave pattern there, since the sine or cosine of a value, as it grows, will output a wave... but straight sinewave functions (like f(x) = sin(x)), look like, well, sinewaves... the plot is a wave that is rounded off at the edges... yours is quite sharp... maybe that's the special thing about it? that it looks like a straight wave?

Actually, that is quite cool, it's a pingpong function... usually when programming anything to have a ping-pong behaviour, you'd need to use boolean logic (if's and then's) to bounce the value back and forth... this function seems to do it mathematically, which, in my coding experience, is always better

Again, here too, pardon any ignorance... I really didn't learn trig in school as I should have... I learned much later by necessity, and I may have a very unusual way of looking at trig stuff...

Cheers

Edit:

Oh, and I think there is a small mistake in the Wolfram Graph you put up... the function reads f = sin(x)...(and so on)... it should be f(x), there is no 'y' parameter anywhere.

Just pointing it out

Edit 2:

Got a little carried away and started playing with that function in Wolfram|Alpha... I gave it the standard sinewave parameters, so it's more adjustable.

like this:
Code:
plot f(x, m, a, p)=  m * asin((cos(ax + p)-sin(ax + p))/sqrt2)

// here, m stands for magnitude, a is the angular frequency, and p is phase

This is actually quite useful

Cheers again

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#### Grover

##### Saturn V Misfire
Donator
well, youve got double angles there, ill have a go at simplifying it before i go back to college in a few minutes, but i dont see how it can be magically special

---------- Post added at 01:27 PM ---------- Previous post was at 01:20 PM ----------

ok, im left with:

sin^-1(1-2sin^2(x)-2cox(x)sin(x))

can anyone take it further?

#### agentgonzo

##### Grounded since '09
well, the cos(2x) - sin(2x) can be simplified down into the form Asin(x + B) for constants A and B that i can't be bothered to work out right now. You can then eliminate the root-2 into the constant A to leave the RHS of the equation being the arcsin of a sin function, which boils down to a linear function - which is what you are getting here.

Given you're going into (and then out of) a trig function that repeats over the input domain, this explains why the output function is repeatable.

sin also has output in the range 0-1 but gives two covers this range twice with its domain of 0-pi, which is why the effectively get the up-down-saw-trangular shape rather than just a repeating linear function with discontinuities.

---------- Post added at 14:33 ---------- Previous post was at 14:29 ----------

well, youve got double angles there, ill have a go at simplifying it before i go back to college in a few minutes, but i dont see how it can be magically special

---------- Post added at 01:27 PM ---------- Previous post was at 01:20 PM ----------

ok, im left with:

sin^-1(1-2sin^2(x)-2cox(x)sin(x))

can anyone take it further?
You're going the wrong way with that. set y = 2x and not worry about the 2 for the moment.
cos + sin = Asin(Bx + C) for some constants A, B and C. Find them. I'll give you a clue, A = sqrt(2), which is why the sqrt(2) on the bottom cancels out.

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#### statickid

##### CatDog from Deimos
Donator
may we ask what the problem was? Perhaps this will shed light on how it can be "special"

#### jthill

##### Member
It demonstrates a consequence of $\frac\Delta{\Delta x}\cos x=-\sin x$ and $\frac\Delta{\Delta x}\sin x=\cos x$.

I think that feeling of special-ness you're getting is your mathematical intuition kicking in, you can feel the basics of calculus before you've been taught. Derivatives are a way of talking about shapes.

#### Grover

##### Saturn V Misfire
Donator
that shows differentiation, i dont see how that becomes involved

#### HarvesteR

##### Member
I think that feeling of special-ness you're getting is your mathematical intuition kicking in, you can feel the basics of calculus before you've been taught. Derivatives are a way of talking about shapes.

That's true. Although for me derivatives were ways of talking about movement :thumbup:

But yeah, there are math concepts which we already have grasped, even though we haven't been formally taught about them (or would be less intimidated by them if we knew that what we think we know is what is being talked about)

Cheers

#### jthill

##### Member
that shows differentiation, i dont see how that becomes involved

I think it's special that sin and cos are each other's derivative (also, irrelevantly, that 0 and $e^x$ are their own, and the russian-doll structure of exponential derivatives), and it answers the only question I can find to ask about what was presented. He's teaching himself so he's looking at it from an unusual perspective, from that viewpoint he sees something interesting about that function. and doesn't know what it is. He's operating in transform space (transform, operate, invert), interesting territory in its own right, so I read his question as "adding a cosine in sine-transform space has a linear effect in the base space?".

Like statickid said, it'd be easier to talk about the geometry here if we could see the geometry that produced the math.

BTW, fireballs619 the Khan Academy videos look pretty darn good, his technique initially seemed to get in the way but over the long haul I think that's not true, it helps you pick your pace. I'm sporadically running the tree for myself, filling in gaps and extending.

I saw Khan's TED talk, when he said teachers have gotten good results inverting the usual teaching structure, assigning his video lectures as homework and using class time for exploring and problem solving, that impressed me. That and the observation from his metrics, about people learning slowly now being an unnervingly bad predictor of how fast they'll learn later. Him pointing that out to a room full of TED attendees was priceless.