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/ |
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