Adittion and double Angle formulae

[Folklore]

Recently been doing some sums to do with the addition formulae, I can;t seem to get the hang of it. has anyone gor any good guides on how to use it correctly etc

the main part I have trouble is when a question states use the Addition fomulae to prove *equation here* especially the double angle formulae

thanks for the help

 

[HallsofIvy]

If you don't show an example, I don't see how we can see where you are going wrong and make suggestions.

If your problem is specifically "use the addition formula to prove the double angle formula", that's almost "mindless".

You are given, say, that sin(x+y)= sin(x)cos(y)+ cos(x)sin(y) and want to derive the formula for sin(x+ y). Isn't letting y= x pretty obvious there?

 

[Folklore]

Yea sorry for been brief there

Example would be using the addition formulae to solve

sin(pi/2) = 1

-------------------------------------

Let me see if I would be correct here

Sin (pi/2) = Sin (90) so I could write

using the formulae Sin (A + B) = Sin A Cos B + Cos A Sin B

Sin (45 + 45) = Sin 45 Cos 45 + Cos 45 Sin 45

--------------------------------------

Please tell me if I'm way off track here, which is highly likley

 

[arildno]

Yea sorry for been brief there

Example would be using the addition formulae to solve

sin(pi/2) = 1

Whatever do you mean by "solve" here?
That is an identity, for the sine function using radian argument.

-------------------------------------

Let me see if I would be correct here

Sin (pi/2) = Sin (90) so I could write

The Sine on the left hand side is a DIFFERENT function than the sine function on the right-hand side, since their argument set is different.

using the formulae Sin (A + B) = Sin A Cos B + Cos A Sin B

Sin (45 + 45) = Sin 45 Cos 45 + Cos 45 Sin 45

--------------------------------------

Please tell me if I'm way off track here, which is highly likley

Yes, you are.

Don't confuse different sine functions, and know what "solve" is, that is learn the differences between definitions, identities and equations.
Definitions are just that, STATEMENTS, neither to be "verified" or "solved" for anything. (They must, however, be understood)
Identities can be verified to hold (for some underlying number set), whereas equations must be solved for those elements in the underlying number sets that makes the equation true. (If all elements in the underlying number set of an equation makes the equation true, then the equation is an identity upon that number set).

 

[Folklore]

Whatever do you mean by "solve" here?
That is an identity, for the sine function using radian argument.

-------------------------------------

The Sine on the left hand side is a DIFFERENT function than the sine function on the right-hand side, since their argument set is different.

Yes, you are.

Don't confuse different sine functions, and know what "solve" is, that is learn the differences between definitions, identities and equations.
Definitions are just that, STATEMENTS, neither to be "verified" or "solved" for anything. (They must, however, be understood)
Identities can be verified to hold (for some underlying number set), whereas equations must be solved for those elements in the underlying number sets that makes the equation true. (If all elements in the underlying number set of an equation makes the equation true, then the equation is an identity upon that number set).

Again I must apologise the exact phrase is prove not solve, I'm sorry there. that does clear a lot up actually. thanks for the help there i'll have to remember the difference in future.

 

[arildno]

You want to "prove" that sin(pi/2) equals 1?
How do you prove that, for the degree sine function Sin(90) equals 1?

That are matters primarily of DEFINITION, somewhat akin to that the symbol number "2" is the name of that quantity you get by adding 1 to itself, i.e, the definition of 2 is 2=1+1.

 

[Folklore]

I guess you can see why I'm confused then in our lecture at university the last question was...

Use the addition formulae to prove that sin(pi/2) = 1 and cos( pi /2 ) = 0.

We don't get the answers to see if we are correct until the next lecture, also our next test will have silimair questions so I want to learn to understand how it is done.

 

[arildno]

Well, what were you given as pre-knowledge?

For example, you certainly can presuppose sin^2+cos^2=1 (*)

If you also were given, say, cos(pi)=-1, you can derive quite a few things:

1. From (*), you get sin(pi)=0
2. Furthermore, from (*) we see that sin^2<=1 for every argument.
Thus, using the cosine double angle formula on cos(pi), we get:
-1=cos^(2)(pi/2)-sin^(2)(pi/2), implying that cos(pi/2)=0

3. Thus, we know that sin(pi/2)=1, or -1.

Perhaps we can tease a bit more out proceeding with this minimal first information, but it would be better if you gave the full assumptions behind the question.

 

[Folklore]

Everything on the section was:

Given the folowing values:

sin(pi/6) = 1/2

cos (pi/6) = Square root 3 / 2

sin(pi/3)= Square root 3 / 2

Cos(pi/3) = 1/2

Use the addition formulae to prove that sin(pi/2) = 1

 

[arildno]

Write: $$\frac{\pi}{2}=\frac{\pi}{3}+\frac{\pi}{6}$$
and proceed.

Notice how crucial it is to convey ALL relevant information, and understand its significance..

 

[Folklore]

Write: $$\frac{\pi}{2}=\frac{\pi}{3}+\frac{\pi}{6}$$
and proceed.

Notice how crucial it is to convey ALL relevant information, and understand its significance..

May I enquire as to how you came to these ?

 

[arildno]

1/2=1/3+1/6, by trivial arithmetic you should be intimately familiar with??

 

[Folklore]

1/2=1/3+1/6, by trivial arithmetic you should be intimately familiar with??

Yes of course I just assumed they where concocted from another formulae.