# Trigonometry to Memorize, and Trigonometry to Derive

• Aug 10th 2011, 01:33 PM
Ackbeet
Trigonometry to Memorize, and Trigonometry to Derive
I have attached a pdf document containing the vast majority of trigonometry I have needed to know on a working basis. The first page consists of trigonometry I think everyone should have memorized. I have never needed to have anything more than the first sheet memorized for any application. The second page is a non-exhaustive sheet of most of the trigonometric identities that I have found useful, and a few more besides.

It is my opinion that a student who memorizes the first sheet, and can derive anything on the second sheet, has a fairly good grasp of trigonometry.

I hope this proves useful.

Attachment 22017
• Aug 10th 2011, 02:18 PM
Siron
Re: Trigonometry to Memorize, and Trigonometry to Derive
I think it's very good document :).
(Do you have more documents for other subjects maybe?)
• Aug 10th 2011, 03:03 PM
Ackbeet
Re: Trigonometry to Memorize, and Trigonometry to Derive
Quote:

Originally Posted by Siron
I think it's very good document :).
(Do you have more documents for other subjects maybe?)

Thank you very much.

I don't really have any others like this one. The reason is the only courses I have taught were Calculus, and this was intended as a review sheet for incoming freshman who were taking my class. I do have problem-solving stickies in the Other Topics and Advanced Applied Math forums. That's about it. Chris L T521 has a very good DE's tutorial, and there's already a LaTeX tutorial and a Calculus tutorial. There's even something in the pre-algebra and algebra forum as well as the linear and abstract algebra forum. So that's most of the forums that I pay the most attention to that even admit of such a document.
• Apr 2nd 2012, 02:35 PM
Ashz
Re: Trigonometry to Memorize, and Trigonometry to Derive
I had an identities quiz, and I used your sheet to help with memorizing the formulas. Thanks a ton! Any pointers on memorizing the unit circle :9
• Jun 30th 2012, 09:33 PM
godfreysown
Re: Trigonometry to Memorize, and Trigonometry to Derive
I find this enormously useful (and encouraging! as I can see how far I've come in a relatively short period of time; and salutary! as I can see how far I have to go before my exam on 17th July)
thanks Ackbeet: thoughtful and very useful!

Godfree
• Jun 30th 2012, 09:57 PM
richard1234
Re: Trigonometry to Memorize, and Trigonometry to Derive
Quote:

Originally Posted by Ashz
I had an identities quiz, and I used your sheet to help with memorizing the formulas. Thanks a ton! Any pointers on memorizing the unit circle :9

You should be able to "visualize" the unit circle, and know which angles correspond to $\displaystyle \frac{\pi}{4}$ or $\displaystyle \frac{\pi}{3}$. Also, by definition, the sine and cosine are the y- and x-coordinates of the point on the circle. No memorization needed...
• Aug 5th 2012, 11:15 PM
louisejane
Re: Trigonometry to Memorize, and Trigonometry to Derive
This really helps! Thank you for sharing your knowledge here. Is it free to ask you anything here that involves mathematics?
• Aug 5th 2012, 11:45 PM
hp12345
Re: Trigonometry to Memorize, and Trigonometry to Derive
Quote:

Originally Posted by louisejane
This really helps! Thank you for sharing your knowledge here. Is it free to ask you anything here that involves mathematics?

Yeah its absolutely free,though you can donate them some money if you think so but its not necessary.
But yeah you cannot use it as a homework completion site ;)
• Aug 21st 2013, 12:03 PM
Seaniboy
Re: Trigonometry to Memorize, and Trigonometry to Derive
Thanks for this helpful document. Much appreciated.
• Jun 5th 2015, 08:50 PM
Archie
Re: Trigonometry to Memorize, and Trigonometry to Derive
I would add to the second page $$\cos{(A-B)} + \cos{(A+B)} = 2\cos A \cos B \\ \cos{(A-B)} - \cos{(A+B)} = 2\sin A \sin B \\ \sin{(A-B)} + \sin{(A+B)} = 2\sin A \cos B$$
• Oct 19th 2015, 12:28 PM
BYUguy
Re: Trigonometry to Memorize, and Trigonometry to Derive
I don't suppose anyone wants to take upon themselves the challenge/effort of writing up how each identity on the 2nd page can be derived from the stuff on the 1st page? (I'd try, but doubt I could get them all).
• Oct 19th 2015, 09:54 PM
JeffM
Re: Trigonometry to Memorize, and Trigonometry to Derive
Quote:

Originally Posted by BYUguy
I don't suppose anyone wants to take upon themselves the challenge/effort of writing up how each identity on the 2nd page can be derived from the stuff on the 1st page? (I'd try, but doubt I could get them all).

You are both missing the point and selling yourself short.

The items on the second page involve virtually no work to derive from the items on the first page. Let's take one of the harder ones.

$tan(x + y) = \dfrac{sin(x + y)}{cos(x + y}.$

That comes from $tan( \theta ) = \dfrac{sin ( \theta )}{cos( \theta )}$ on page 1.

$So\ tan(x + y) = \dfrac{sin(x)cos(y) + cos(x)sin(y)}{cos(x + y)}$

That comes from $sin(x + y) = sin(x)cos(y) + cos(x)sin(y)$ on page 1.

$So\ tan(x + y) = \dfrac{sin(x)cos(y) + cos(x)sin(y)}{cos(x)cos(y) - sin(x)sin(y)}.$

That comes from $cos(x + y) = cos(x)cos(y) - sin(x)sin(y)$ on page 1.

Now translate into tangents using $tan( \theta ) = \dfrac{sin( \theta )}{cos( \theta )}$ from page 1.

$So\ tan(x + y) = \dfrac{cos(x)cos(y)\left \{\dfrac{sin(x)}{cos(x)} + \dfrac{sin(y)}{cos(y)}\right \}}{cos(x)cos(y)\left \{1 - \dfrac{sin(x)}{cos(x)} * \dfrac{sin(y)}{cos(y)}\right \}} \implies$

$tan(x + y) = \dfrac{tan(x) + tan(y)}{1 - tan(x)tan(y)}.$

Now this is just algebra applied to a few basic memorized formulas. Each of the formulas on the second page involve a few simple manipulations of what is on the first page. You can memorize the second page, but you do not need to. Furthermore, deriving them on your own will give you confidence in your ability to handle more complicated transformations among trigonometric functions.
• Feb 23rd 2017, 09:35 AM
jpompey
Re: Trigonometry to Memorize, and Trigonometry to Derive
This is very helpful, thank you.