Formulas for Sine and Cosine
Date: 1/24/96 at 21:23:51 Subject: What are the definitions of sin, cos, and tan in terms of theta? From: Michael Allman Dr. Math, I have been searching for an answer for several days and am really stuck. Even my precalc. teacher doesn't know (by the way - I am a high school junior.) What are the definitions of sin, cos, and tan in terms of theta? So far, I have come upon a definition that looks something like: sin x = x + x^3/3! - x^5/5! + x^7/7! . . . but I do not know how to handle this series. Also, because of the ambiguous use of the variable x, I do not even know if this series is the answer I'm looking for. So, is there a nice neat answer? I also have a question about the notation of the trig. functions. If sin is a function and f(x) describes a function of x (e.g. f(x) = 2x), why is sin theta defined by y and r in my textbook? It seems to me that sin theta should be defined by theta and that sin (y,r) should be defined by y and r. In a nutshell: 1. What is the definition of sin, cos, and tan in terms of the angle (not in terms of x, y, or r?) 2. Why is the function sin theta defined by y and r, and not by theta? Thank you for your help! Michael Allman
Date: 1/27/96 at 17:3:42 From: Doctor Ken Subject: Re: What are the definitions of sin, cos, and tan in terms... Hello! The way Sine and Cosine are defined is usually in terms of the unit circle. What you do is draw a circle of radius 1, whose center is at the point (0,0). Then you draw a ray coming out from the origin that makes an angle of theta with the x-axis. Note that the way you measure this angle is by starting at the x-axis, and travelling COUNTER- CLOCKWISE until you hit the ray in question. Thus this is about 60 degrees: / / / / / theta /_________________> x-axis And this is about -60 degrees: __________________> x-axis \ theta \ \ \ \ \ This ray will intersect with the unit circle at a point (x,y). Cos(theta) is defined as x, the first coordinate of this intersection point, and Sin(theta) is defined as y, the second coordinate. Then Tangent is defined as Sin/Cos, and so on. >2. Why is the function sin theta defined by y and r, and not by theta? Well, I'm not really sure what you mean by y and r. What are your y and r? -Doctor Ken, The Geometry Forum
Date: 1/29/96 at 15:1:2 Subject: Re: What are the definitions of sin, cos, and tan in terms of From: Michael Allman Dr. Math, You misunderstood my two questions. The first one asked for the definitions of the trigonometric functions in terms of theta, theta being the angle in question. So if I needed to know the value of sin 54.2 degrees and I did not have a calculator, how would I calculate this? When you wrote "Cos(theta) is defined as x, the first coordinate of this intersection point, and Sin(theta) is defined as y,...", you defined cos and sin in terms of the coordinates x and y. As for my second question, y stands for the y coordinate of a point in a plane, and r stands for the length of the segment between that same point and the origin of the coordinate system. See below. y-axis (x,y) | / | / | r / | / | / | / | / | / _____|/______________________ x-axis origin | | | So, since sin theta is a function of theta, why is it defined by (y/r) and not by theta itself? This seems to be an incongruity in function notation. I hope this is clearer. Thank you. Michael Allman
Date: 2/6/96 at 15:13:36 From: Doctor Ken Subject: Re: What are the definitions of sin, cos, and tan in terms of theta? et al. Hello! Thanks, that is clearer. First I should tell you that I think you don't actually mean "definition." There is only one definition of Sine and Cosine of an angle, and that's what I gave you. It sounds like what you're looking for is more of a "formula" for the Trig functions in terms of the angle, right? In that case, the first few terms of the series you mentioned will work fine. So for sin(x), a pretty good approximation will be x - x^3/6 + x^5/120, and for cos(x) we have 1 - x^2/2 + x^4/24 - x^6/720. If you don't need to be so accurate, you can drop the last term on each of those series. Another thing is that if you're going to try to take the Sine or Cosine of a theta that's not very close to 0, you should do some conversion first to improve your accuracy (these formulae are most accurate around 0). So if you were going to take the Cosine of 17Pi + 2.8, you'd say, "hey, that's about the same thing as 18Pi - .3415, which will have the same Sine as if theta were -.3415"; this gives you a MUCH better estimate. In general, you want to get your angle between -Pi/2 and Pi/2. Another thing to keep in mind: all you'll ever be able to find is an approximation like this. For most angles (all except the special ones like Pi/3 and stuff) there is no neat, closed form expression for its Sine in terms of the original angle measure. As for your second question, rest assured that Sine is well-defined. Each value of theta gives a unique point of intersection with the unit circle. Sure, it's defined in terms of y/r (in your notation), but the y and the r are determined by theta. It seems to me that you're confusing the notion of "definition" and "formula"; if you were expecting the definition of Sine to be a bunch of square roots and plusses and stuff, all in terms of theta, then that would explain your confusion. But a definition can be whatever it wants to be: in terms of theta, or y, or r, or the color of the sky. Hope this helps. -Doctor Ken, The Math Forum
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