Date: Sat, 5 Nov 1994 19:07:11 +0000 From: Anonymous I just read about your service and had to send you a message. I am a junior in High School and currently taking a pre-calc. course. We started the year with trig. identities and I have never covered this material before. What confuses me is how to find the exact value (i.e., not a decimal answer) of a angle(in radians or degrees) QUICKLY. For several of our problems we need to know almost instantaneously the value of say, sec 15, sin 45, cos 60, etc. Instead of drawing a triangle on a cortesian plane and using the 30x30x60 or 45x45x90 rules is there an easier way to derive the answers to these questions quickly? Thanks in advance for any help, Anonymous Thank you.
X-Sender: firstname.lastname@example.org Date: Sun, 6 Nov 1994 15:58:51 -0500 Hi there! Thanks for writing to Dr. Math! Ah, trig identities. Those were the days. Pop quizzes every Friday on the precise definition of cos(x/2). If there was a really easy way to do what you want to do, then there would be a really easy way to get an A in trig, which of course, there isn't. I always found the triangle method to be kind of tedious. What I *did* find to be useful was to try to memorize the following: sin 30 = 1/2 sin 45 = root(2)/2 cos 30 = root(3)/2 If you know these, then you're on your way to being able to find most of the answers you're looking for. Remember a few facts. First of all, cosx = sin (90-x). In other words, sin 30 = cos 60, sin 60 = cos 30, sin 45 = sin 45. Therefore, if the question were to ask for cos 60, what I would do is start with sin 30, equate that with cos 60, and have the answer. To find tangents, I always found it easier to manipulate sin and cos than to memorize the actual values. Therefore, if you need to find tangent 30, you know that tan = sin/cos, that sin 30 = 1/2 and cos 30 = root(3)/2, and therefore tan 30 = 1/root(3) or root(3)/3. To deal with cosecant, secant, and cotangent, just remember that they are the multiplicative inverses of the trigonometric functions we talked about above. So cotangent 30 = 1/tan 30 = root(3). Finally, how to deal with stuff like sec (15). Here, you have to get used to using all those identities you'll find in your trig book. You'll have to memorize a few of them, and maybe learn to derive a few of them on the spot if needed like above (i.e. learn cos x/2 and sin x/2, and then get tan x/2 by dividing the two). There is no easier way to get these difficult answers. In short, there will probably be a grid in your math book detailing the important trigonometric values (i.e. sin, cos, tan, csc, sec, cot) for 30, 45 and 60. Get to know the left part of this grid, particularly the sin and cos as described above. Then, starting from that base, you'll begin to be able to manipulate things in your mind. This is what I did, and I think it helped (I got used to trig a lot faster perhaps than if I had used the triangle method) Furthermore, don't be afraid to draw little cos and sin graphs on the side of your paper for confirmation of what you think the values are. And finally, don't worry! As you use trig more throughout the year, this stuff will begin to be second-hand. In the beginning, the above worked for me. I hope it eventually makes things a little easier for you. Sincerely, Phil, Dr. Math
Date: Sun, 6 Nov 1994 16:00:55 +0000 From: Anonymous I really appreciated your reply about trig identities. What you said makes sense and I printed it out so I might show it to my friends who are also a little confused right now. I heard about the Dr. Math service on Usenet, and read that it was NSF-funded. It really seems like a super service, you should be commended. I really love math, I just don't enjoy learning a theorem of method without seeing the background behind it. This is the method my teacher seems to favor. Thanks again, Anonymous
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.