# Need advice on how to study trigonometry

1. Oct 26, 2009

### iamsmooth

I'm currently in first year calculus, and I'm having trouble remembering all my trig for use in derivatives.

I have some stuff memorized, obviously soh cah toa, and the derivatives of the six trig functions.

i.e. derivatives of
sinx = cosx
cosx = -sinx
tan = sec^2(x)

etc...

but when using them in an equation there are other properties which I'm not sure of. For example: cos(0) = 1 (or something like that?) and sinx/x = 1. No where in my trig section does it point out these properties or explain them (I have,however, found proofs of sinx/x = 1 online), it seems to be required knowledge that the textbook assumes you know... but I don't.

So what I'm asking is, does anyone know where I could look (possibly a webpage) where I could find these properties? Or is there a proper name I can check the appendix of my textbooks for that might explain these things for me.

2. Oct 26, 2009

### pbandjay

A great thing to know is the unit circle:

http://en.wikipedia.org/wiki/Unit_circle

This is a circle of radius one, and in each ordered pair, the x-component represents cosine while the y-component represents sine. If that isn't very clear, just visualize a right triangle from the origin to that point. Only memorizing a couple numbers is needed to be able to figure out the sine and cosine of nice angles.

Also, don't forget the limit in your expression: "sinx/x = 1"...

$$\lim_{x\to0}\frac{\sin x}{x}=1$$

as (sin x)/x does not always equal one, for example:

$$\frac{\sin(\pi/4)}{\pi/4}=\frac{2\sqrt{2}}{\pi}\neq1$$

Last edited: Oct 26, 2009
3. Oct 26, 2009

### lurflurf

These are the most important ones
lim_x->0 sin(x)/x=1
sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
cos(x+y)=cos(x)cos(y)-sin(x)sin(y)
cos(x)cos(x)+sin(x)sin(x)=1
tan(x)=sin(x)/cos(x)
sec(x)=1/cos(x)
csc(x)=1/sin(x)

Proofs require defiinitions to works from, for example the above formulas in addition to summarizing the basic properties constitute definitions. Less to memerize!
For example it is more sensible to include lim_h->0 sin(h)/h=1 in the definition than try to prove it because the first step of the prooff would be lim_h->0 sin(h)/h=some number and we want that number to be unity. You can memorize the other formulas, but most important is to remenber they exist. So for example
sin(x)-sin(y)=2cos((x+y)/2)sin((x-y)/2) is nice to remember, but if you remember that there exist something like that it is easy to derive or look up. One good thing to do is solve many problems here are a few things that have come up in threads here lately.

$$\frac{\sin(x)}{\cos^2(x)+\cos(x)-2}=\frac{-\cos\left(\frac{x}{2}\right)}{3\sin\left(\frac{x}{ 2}\right)-2\sin^3\left(\frac{x}{2}\right)}$$

sin(2x + pi/3) = sin(2x) + sin(2(x+pi/3))

cos(theta/2)=$$\sqrt{\frac{1+cos\theta}{2}}$$

sin(x)+cos(x)=sqrt(2)sin(x+pi/4)

cos(x)-sin(x)=2sin(pi/4)cos(x+pi/4)

the above two can be used to easily find min and max of cos(x)-sin(x) and cos(x)+sin(x) without using calculus

http://en.wikipedia.org/wiki/List_of_trigonometric_identities

Last edited: Oct 26, 2009