# Adittion and double Angle formulae

• Folklore
In summary, the conversation is about using the addition formulae to prove identities involving the sine and cosine functions. The main issue is understanding the difference between definitions, identities and equations. The conversation also provides an example of using the addition formulae to solve for sin(pi/2) and how crucial it is to convey all relevant information and understand its significance. It is also mentioned that the concept may be studied in a university lecture, and the conversation concludes with a question about how the previous values were derived.
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

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?

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

Folklore said:
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).

arildno said:
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.

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.

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.

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.

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

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..

arildno said:
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 ?

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

arildno said:
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.

## 1. What are addition formulae for trigonometric functions?

Addition formulae for trigonometric functions are mathematical equations that allow us to find the trigonometric function of the sum or difference of two angles, given the trigonometric functions of the individual angles. These include the sum and difference identities for sine, cosine, and tangent functions.

## 2. How do I use addition formulae to simplify trigonometric expressions?

By using addition formulae, we can simplify trigonometric expressions by replacing the sum or difference of two angles with a single trigonometric function. This can help us solve equations, evaluate trigonometric functions at specific values, and prove trigonometric identities.

## 3. What are double angle formulae for trigonometric functions?

Double angle formulae for trigonometric functions are equations that allow us to find the trigonometric function of an angle that is twice the size of a given angle. These include the double angle identities for sine, cosine, and tangent functions.

## 4. How do I use double angle formulae to solve trigonometric equations?

By using double angle formulae, we can solve trigonometric equations by replacing the double angle with a single trigonometric function. This can help us find exact solutions to equations that involve trigonometric functions.

## 5. Can I use addition and double angle formulae for all trigonometric functions?

Yes, addition and double angle formulae can be used for all six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). However, for the cosecant, secant, and cotangent functions, we use the reciprocal identities of the sine, cosine, and tangent addition and double angle formulae.

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