# Trig Identities

1. Nov 17, 2011

### sandy.bridge

Hello all,
I'm wanting to learn how to derive all of the trig identities (well, not all, but the most common) rather than memorizing them. Perhaps someone here could provide me with a list of "essentials" that are the framework for deriving others. For example, I know there are a few that can be derived from sin^2x+cos^2x=1. What else?

2. Nov 18, 2011

### lurflurf

Different "essentials" can be stated one possibility is

1)sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
2)cos(x+y)=cos(x)cos(y)-sin(x)sin(y)
3)sin2(x)+cos2(x)=1
4)sin'(0)=1

alternatives to 4) are x~sin(x) for small x and various inequalities like cos(x)<sin(x)/x<1

These are not only enough to derive usual identities, but to define sine and cosine.
Other functions like secant and cotangent are defined as quotients of sine and cosine.

3. Nov 18, 2011

### sandy.bridge

Sweet, thanks!

4. Nov 18, 2011

### I like Serena

Hi sandy.bridge!

The trig identities can be derived from Euler's formula.
See "[URL [Broken]

Euler's formula comes in 3 forms:
$$e^{ix} = \cos x + i \sin x$$$$\cos x = {1 \over 2}(e^{i x} + e^{-i x})$$$$\sin x = {1 \over 2i}(e^{i x} - e^{-i x})$$

For instance:
$$\cos 2x = {1 \over 2}(e^{i 2x} + e^{-i 2x}) = {1 \over 2}((e^{i x} + e^{-i x})^2 - 2 e^{i x} e^{-i x}) = 2 \cos^2 x - 1$$

Last edited by a moderator: May 5, 2017