Trig Identities

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: pspecto1@cc.swarthmore.edu
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