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Trig in the Age of Calculators, and Why Don't We Just Measure the Triangles Directly?

Date: 01/10/2015 at 14:31:01
From: 
Subject: Sine, Cosine and Tan functions 

Hey! I'm having trouble understanding trig. I was fine with it till 
yesterday. Then we had a partnered assignment and my partner kind of 
confused me. 

I have a couple questions. 

1.) Are sine, cosine and tangent all functions that are applied to an 
angle which give the same answer as if you were to take the sides' ratios? 
 
2.) Why do we write SinA or CosA or TanA? This is more of a general 
question, I guess; but if we know the value of the angle, why don't we 
just solve the side with CosA or TanA or SinA? 

For example,

   Cos(82) = 0.94967769788

Why don't we solve Cos(82) as well? After all, we know that this side is 
the a/h ratio.

I understand what the inverse operations do and I understand the sine and 
cosine laws, but I'm having trouble getting my head around what sine and 
cosine and tangent *are*. We keep being told they're ratios, but I've read 
that they're functions ... so I'm assuming that the function is equal to 
the ratio? and if you apply the function to an angle, it doesn't matter if 
you know the sides or not, you can still get the sides' ratios? 

I'm really confused. I don't understand if to get the sine of something 
you do o/h, or if sine is a function and you input an angle and it gives 
you the same answer as if you were to take the opposite and hypotenuse of 
that angle and find their ratio. 

I think I need a slightly more detailed explanation of the trig ratios. 



Date: 01/10/2015 at 17:54:12
From: Doctor Peterson
Subject: Re: Sine, Cosine and Tan functions 

Hi, Emily.

I imagine a lot of students have questions like yours when they first 
learn trig, but they don't ask them! Let's get them cleared up.

Each trig function is defined as one of the ratios you would get if you 
made a right triangle using the given angle. (I'll talk a little later 
about how to improve this definition, but I'm starting where you 
presumably are.)

So given any acute angle, we are defining a FUNCTION based on a triangle 
we haven't actually made. 

The first big question to ask is, are you sure this really defines a 
function? That is, for any input (angle) can we be sure that there is only 
one output value?

Let's focus on the sine function, to be specific. If we're given an angle, 
in order to find the tangent of that angle by the definition, we need to 
make a right triangle. 

But which one? We could make infinitely many of them:

                 /
               +
             / :
           +   :
         / :   :
       /   :   :
     /     :   :
   +-------+---+---

Those are two of the possible triangles. The important thing is that the 
ratio of opposite to hypotenuse is the same for ALL of them, because they 
are all similar triangles. It doesn't matter how long we make the sides, 
provided we have the right angle. So given an angle, you'll always get the 
same value for its sine.

The second question to ask is, how can we find the value of the function? 
We can't actually draw the triangle whenever we need to calculate sin(A), 
and that wouldn't be very accurate anyway. In the VERY old days (the 
ancient Greeks and maybe the Babylonians), they made tables of trig 
functions, probably by using some of the identities you will be learning, 
to derive the sine of a new angle from the sines they'd already 
calculated. 

You can find the sine of twice an angle using only the sine (and cosine) 
of the angle itself. And it turns out that the sine of a VERY small angle 
is very close to the radian measure of that angle (which, if you haven't 
learned, is the arc length of the angle on a unit circle: A*pi/180). There 
are also a couple special angles for which we know the exact value of the 
trig functions by geometry; we can start with those angles and work down 
to smaller angles.

Later table-makers used other methods that have been discovered over the 
years. For example, you can get a more accurate value for the sine of a 
small angle by using a few terms of this series (where x is the radian 
measure of the angle, and 3!, for example, means 3*2*1 = 6):

   sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

There are even quicker formulas, but they would be harder to describe.

In MY "old days," we looked up sines in such a table; today, our 
calculators do for us calculations similar to those done by the 
table-makers. But as far as we are concerned, we can imagine there is a 
genie in there drawing a really accurate triangle and measuring its sides 
for us. It gives the same result.

You may find this interesting:

  Methods of Computing Trig Functions
    http://mathforum.org/library/drmath/view/52576.html 

So the answer to most of your questions is "yes." The trig functions are 
functions (that are evaluated in complicated ways we don't need to know) 
that give the value of the appropriate ratio as if we had drawn the 
triangle. It is a function the value of which is defined by a ratio, but 
not usually evaluated using that ratio. Once you've evaluated the 
function, you can use it to find what the sides of a particular triangle 
would be much more accurately than if you measured them.

I'm not sure whether I've answered all your questions, especially the 
second; feel free to write back and ask more until we get it all clear.

Oh, I said I'd talk about a better definition. The definition in terms of 
right triangles applies only to acute angles, so we extend it further. 
One way to do that is to use a unit circle (radius 1) and say that, for 
example, the sine of any angle is the y-coordinate of the point where the 
terminal ray in standard position intersects the circle. For an acute 
angle, you can draw a triangle and see that this definition is identical 
to yours. But now we can use any angle at all. This makes trig much more 
useful. 

But you'll be getting there, if you haven't already.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Trigonometry

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