Learning Trig Identities: Deriving the Essentials

[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?

 

[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.

 

[sandy.bridge]

Sweet, thanks!

 

[I like Serena]

Hi sandy.bridge!

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

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

 

[CellCoree]

I fully support your approach to learning trigonometric identities by deriving them instead of just memorizing them. This not only helps with understanding the concepts better, but also allows for a deeper understanding of how these identities are interconnected.

Some other essential identities that can be derived from the fundamental identity of sin^2x+cos^2x=1 are:

1. Double angle identities: These include sin2x=2sinxcosx, cos2x=cos^2x-sin^2x, and tan2x=2tanx/1-tan^2x.

2. Half angle identities: These are derived by using the double angle identities and the Pythagorean identities. For example, sin(x/2)=±√[(1-cosx)/2].

3. Sum and difference identities: These include sin(x+y)=sinxcosy+cosxsiny, cos(x+y)=cosxcosy-sinxsiny, and tan(x+y)=(tanx+tany)/(1-tanxtany).

4. Product to sum identities: These are derived using the sum and difference identities and the double angle identities. For example, sinxcosy=(1/2)(sin(x+y)+sin(x-y)).

5. Pythagorean identities: In addition to the fundamental identity, there are two other Pythagorean identities that can be derived: tan^2x+1=sec^2x and cot^2x+1=csc^2x.

These are just some of the essential identities that serve as the building blocks for deriving others. I encourage you to continue exploring and deriving more identities, as it will greatly enhance your understanding and mastery of trigonometry. Best of luck in your learning journey!