Math Is Fun Forum

numquester

Member

Registered: 2016-12-12

Posts: 19

Why are trig functions taught as being blackboxes?

Why is trigonometry taught from the perspective of trig functions being blackboxes?  It seems to be counter to what math is about.

Given some function, f(x) = some calculation involving x, we see what the function is and does.  For example, given the function, f(x) = x^2, we see that the function squares x.

But given a trig function, for example, sin(angle) = opposite/hypotenuse, we don't actually see what the function is and does.  sin() is a blackbox, and we don't see what it does to the angle.  And possibly creating much confusion here is that the notation used for trig functions can make it appear that, opposite/hypotenuse, is the operation of the function.

Last edited by numquester (2018-03-01 07:36:53)

numquester

Member

Registered: 2016-12-12

Posts: 19

Re: Why are trig functions taught as being blackboxes?

Expanding on the above post.

For some function, f(x) = x^2, for example:

f(x), is a label for a function.  It is not the operations of a function.
x^2, is the operation of the function.  It is what the function does.

For the sin function, sin(angle) = opposite/hypotenuse:

sin(angle), is also only a label for a function.  Is is not the operations of the function.
opposite/hypotenuse, is a ratio (a number) that is equivalent to the result of the operations of the sin function.
The actual operations of the sin function are not represented in the notation.

But both of the above are notated as if having the same general form:  A label and operation(s).

f(x) = x^2
sin(angle) = opposite/hypotenuse

And at least for all the trig resources which I have seen, a distinction is never made that different things are being represented by the same notation form.  That seems like a disaster in waiting for students of trig.

Last edited by numquester (2018-03-01 07:37:55)

numquester

Member

Registered: 2016-12-12

Posts: 19

Re: Why are trig functions taught as being blackboxes?

Possibly causing even more confusion, we might see this:

f(x) = sin(x)

What we have in that notation is nothing but one label being equivalent to another label.  No operations are present.

Bob

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Registered: 2010-06-20

Posts: 10,626

Re: Why are trig functions taught as being blackboxes?

hi numquester

In maths a black box like that is called a function.  A number goes in and a number comes out.  'squared', sin and log are all examples.  There is a formula using x for sin but I don't think you'll like it:

This is a series of terms going on for ever following that pattern.  With a restriction on x it is possible to evaluate sin(x) by using that formula and that is roughly how calculators do it.  It's one of a number of power series.  There's a similar one for cos(x).

If you had to re-invent sin(x) from scratch you could do it by drawing right angled triangles and carefully measuring the opposite and hypotenuse and doing the calculation opp/hyp.  For a given angle you'd get a number for sin(x) so it is still a black box and that fraction (ratio) instructs how to evaluate it.

You've got used to evaluating the 'square' function because you only have to know how to multiply to do the function.  That is clearly so straight forward for you that you haven't realised you are doing a black box in that case too. sin is just a more complicated function.

Hope that helps,

Bob

7! pronounced factorial 7 is 7x6x5x4x3x2x1

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

numquester

Member

Registered: 2016-12-12

Posts: 19

Re: Why are trig functions taught as being blackboxes?

Apologies for not replying after you gave that explanation.  It is a good explanation.  Thanks.