Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

C3 Trig - Compound angles proof

https://onedrive.live.com/redir?resid=FA1A3849E1FA1C5D!5311&authkey=!ALyFbBZ_hZbV0bE&v=3&ithint=photo%2cJPG

I don't get how you get from the second last line to the last line.

Can someone explain?

It's all about cancellation :smile:

:smile:

Remember that

\dfrac{a}{b} \times \dfrac{b}{c} = \dfrac{a}{c}

Not a fan of geometric proofs. Prefer using the fact that

\displaystyle \frac{1}{2}(e^{ix}-e^{-ix})=\sin x

and

\displaystyle \frac{1}{2}(e^{ix}+e^{-ix})=\cos x

and then do it algebraically.

Original post by B_9710

Not a fan of geometric proofs. Prefer using the fact that

\displaystyle \frac{1}{2}(e^{ix}-e^{-ix})=\sin x

and

\displaystyle \frac{1}{2}(e^{ix}+e^{-ix})=\cos x

and then do it algebraically.

But then your point is invalid because those equations rely on the compound angle formula.

Original post by MathsAndChess

But then your point is invalid because those equations rely on the compound angle formula.

No they don't.

Original post by B_9710

No they don't.

How do you prove that statement then?

Original post by MathsAndChess

How do you prove that statement then?

Complex numbers.

Original post by B_9710

Complex numbers.

I'm amusing you mean though the fact

\exp(i\theta) = \cos \theta + i\sin \theta

username1560589

Original post by MathsAndChess

I'm amusing you mean though the fact

\exp(i\theta) = \cos \theta + i\sin \theta

Depends how you define sin and cos.

I guess if you're speaking from a geometric definition of sin and cos. Which I am then my statement is true. Else then I see why you wouldn't require this.

Original post by MathsAndChess

I guess if you're speaking from a geometric definition of sin and cos. Which I am then my statement is true. Else then I see why you wouldn't require this.

If you use the definitions above and find

\sin x \cos y +\cos x \sin y

you will find that by the definition above that this is the same as

\sin(x+y) .

You can find other results similarly.

username1560589

Original post by MathsAndChess

I guess if you're speaking from a geometric definition of sin and cos. Which I am then my statement is true. Else then I see why you wouldn't require this.

Yeah if you use the geometric definitions you need to derive the double angle formulae before you can do anything interesting.

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