Why does 2(sin(a)*cos(a))= sin(2a) ?

[Georgepowell]

I think it does... have I got that right?

If I have then why does it? is there a simple explanation?

 

[mgb_phys]

Yes it does.
It's easy to prove from the sum formula, sin (a + b)=sin a cos b + cos a sin b
Of course now you have to prove that!

 

[Georgepowell]

Yes it does.
It's easy to prove from the sum formula, sin (a + b)=sin a cos b + cos a sin b
Of course now you have to prove that!

Go on...

How would I do that? I don't need a solid proof, just a simple explanation would do...

Thanks

 

[D H]

Set b=a.

 

[Georgepowell]

Set b=a.

Yeh I got that part, What I wanted was for someone to tell me why

sin(a + b) = sin(a)cos(b) + sin(b)cos(a) ?

 

[D H]

One way to derive the sine and cosine of the sum (or difference) of pair of angles is via Euler's identity:

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

Setting \theta=a+b,

\aligned<br /> \exp(i(a+b)) &amp;= \exp(ia)\exp(ib) \\<br /> &amp; = (\cos a + i\sin a)(\cos b + i\sin b) \\<br /> &amp;= (\cos a \cos b - \sin a \sin b) + i(\sin a \cos b + \cos a \sin b)) \\<br /> &amp;= \cos(a+b) + i\sin(a+b)<br /> \endaligned

Equating real and imaginary parts,

\aligned<br /> \cos(a+b) &amp;= \cos a \cos b - \sin a \sin b \\<br /> \sin(a+b) &amp;= \sin a \cos b + \cos a \sin b<br /> \endaligned

 

[Georgepowell]

\aligned<br /> &amp;= (\cos a \cos b - \sin a \sin b) + i(\sin a \cos b + \cos a \sin b)) \\<br /> &amp;= \cos(a+b) + i\sin(a+b)<br /> \endaligned

Can you go over this step please? Thanks

 

[nicksauce]

Can you go over this step please? Thanks

By using e^{i\theta}=\cos{\theta} + i\sin{\theta}
the LHS becomes
\cos{a+b} + i\sin{a+b}.

 

[Georgepowell]

By using e^{i\theta}=\cos{\theta} + i\sin{\theta}
the LHS becomes
\cos{a+b} + i\sin{a+b}.

haha, ok... I think this is getting well out of my depth. Unless there is a more simple explanation, I'm outahere.

Anyway - Thanks for all your help everyone.

 

[HallsofIvy]

Assuming that you are using the "circle" definition of sine and cosine: that the coordinates of a point a distance t along the circumference of the unit circle from (1, 0) are (cos(t), sin(t)), then you can use argue that the straight line distance from (1, 0) to (cos(t), sin(t)) and from (cos(s), sin(s)) to (cos(s+t), sin(s+t)) are the same because because they both cut an arc of length t.
\sqrt{(cos(t)- 1)^2+ sin^2(t)}= \sqrt{(cos(s)-cos(s+t))^2+ (sin(s)- sin(s+t))^2}
Squaring both sides and multiplying everything out,
cos^2(t)- 2cos(t)+ 1+ sin^2(t)= cos^2(t)- 2cos(s)cos(s+t)+ cos^2(s+t)+ sin^2(s)- 2sin(s)sin(s+t)+ sin^2(s+ t)
The cos^2(t) and sin^2(t) on each side cancel and cos^2(s+t)+ sin^2(s+ t)= 1 which cancels the "1" on the left. That leaves
-2cos(t)= -2 cos(s)cos(s+t)- 2sin(s)sin(s+t) so cos(t)= cos(s)cos(s+t)+ sin(s)sin(s+t).

Let a= s, b= s+t. Then t= b- a and we have cos(b-a)= cos(a)cos(b)+ sin(a)sin(b).

Replacing a by -a and using the fact that cos(-a)= cos(a) while sin(-a)= -sin(a), we have cos(a+ b)= cos(a)cos(b)- sin(a)sin(b).

Use the fact that sin(x)= cos(\pi/2- x) to change to a formula for sin(a+b).

That's the proof that is used in most pre-calculus classes.

 

[Georgepowell]

...

That's the proof that is used in most pre-calculus classes.

Thanks, that's what I wanted. I understand it now.

I don't like memorising formulas without understanding them, and my teachers teach the formulas and not the explanations. Which I don't like.

I suppose I'm not even meant to know this formula yet, but I'm interested.

 

[tiny-tim]

Hi Georgepowell!

Draw a right-angled triangles PQR and PQS with R = S = 90º and QPR = A + B and QPS = A (so SPR = SQR = B).
Draw the perpendiculars from S to PR and QR.

Then, if PQ = 1, QR = sin(A+B) …

and you can work out the rest!

 

[Georgepowell]

Hi Georgepowell!

Draw a right-angled triangles PQR and PQS with R = S = 90º and QPR = A + B and QPS = A (so SPR = SQR = B).
Draw the perpendiculars from S to PR and QR.

Then, if PQ = 1, QR = sin(A+B) …

and you can work out the rest!

Is this what you meant?

 

[HallsofIvy]

But how do you apply that, say, to A= 50 degrees, B= 60 degrees?

 

[tiny-tim]

Is this what you meant?

Yes … and now draw the perpendiculars from S to PR and QR

(and note that the first one will be parallel to QR).

(and the circle isn't needed)

 

[maze]

A braindead way to prove any trig identity is to write the identity in terms of complex exponentials and simplify as much as possible until 0=0 is reached. Then the reverse of your steps is the proof of the identity.

For example,
2 sin(a) cos(a) - sin(2a) = 0

2\left(\frac{e^{i a}-e^{-i a}}{2 i}\right)\left(\frac{e^{i a}+e^{-i a}}{2}\right) - \frac{e^{i 2a}-e^{-i 2a}}{2 i} = 0

\frac{1}{2 i}\left(e^{i a}e^{i a} + e^{i a}e^{-i a} - e^{-i a}e^{i a} - e^{-i a}e^{-i a}\right) - \frac{1}{2 i}\left(e^{i 2a}-e^{-i 2a}\right) = 0

\frac{1}{2 i}\left(e^{i 2a} - e^{-i 2a}\right) - \frac{1}{2 i}\left(e^{i 2a}-e^{-i 2a}\right) = 0

0 = 0

Now a proper proof would run these steps in reverse.

 

[HallsofIvy]

A braindead way to prove any trig identity is to write the identity in terms of complex exponentials and simplify as much as possible until 0=0 is reached. Then the reverse of your steps is the proof of the identity.

For example,
2 sin(a) cos(a) - sin(2a) = 0

2\left(\frac{e^{i a}-e^{-i a}}{2 i}\right)\left(\frac{e^{i a}+e^{-i a}}{2}\right) - \frac{e^{i 2a}-e^{-i 2a}}{2 i} = 0

\frac{1}{2 i}\left(e^{i a}e^{i a} + e^{i a}e^{-i a} - e^{-i a}e^{i a} - e^{-i a}e^{-i a}\right) - \frac{1}{2 i}\left(e^{i 2a}-e^{-i 2a}\right) = 0

\frac{1}{2 i}\left(e^{i 2a} - e^{-i 2a}\right) - \frac{1}{2 i}\left(e^{i 2a}-e^{-i 2a}\right) = 0

0 = 0

Now a proper proof would run these steps in reverse.

I believe that was what mbg_phys did in the first several responses but GeorgePowell was not familiar with the exponential form of sine and cosine and could not follow.

 

[tiny-tim]

keep it real

I believe that was what mbg_phys did in the first several responses but GeorgePowell was not familiar with the exponential form of sine and cosine and could not follow.

Yeah … keep it real!

Support CAMREG …

the CAMpaign for REal Geometry! ​

 

[Georgepowell]

Yes … and now draw the perpendiculars from S to PR and QR

(and note that the first one will be parallel to QR).

(and the circle isn't needed)

That's a good simple proof that I understand, but doesn't it only prove it for:

2A + B = 90 ?

**note: In the picture, I changed the letters, but it is the same concept. And the circle was just to help me get the right-angles right**

 

[tiny-tim]

That's a good simple proof that I understand, but doesn't it only prove it for:

2A + B = 90 ?

Hi Georgepowell!

(can't see your new diagram yet)

No, I think that it works for any A + B < 90º …

and even for A + B ≥ 90º provided both A and B < 90º if you put R on the other side of the circle.

**note: In the picture, I changed the letters, but it is the same concept. And the circle was just to help me get the right-angles right**

Yeah, the letters don't matter …

it's always best to use whatever letters you find easiest.

hmm … now you mention it, my own diagram (which you can't see would have been a lot better if i'd started off with a circle )​

 

[maze]

I believe that was what mbg_phys did in the first several responses but GeorgePowell was not familiar with the exponential form of sine and cosine and could not follow.

Err, not really. The other one was a little trickery using Euler's formula, a method which does not generalize to prove all other identities.

 

[tiny-tim]

support CAMREG!

Hi Georgepowell!

(can't see your new diagram yet)

ooh, i can see your diagram now …

very nice

mmm … if you've nothing better to do,
try proving cos3x = cos³x - 3.cosx.sin²x with a diagram ​

EDIT: Just noticed … the angle PQO in the diagram is wrongly labelled as A (instead of 90º - A - B), … but it doesn't affect the proof.

 

[D H]

Err, not really. The other one was a little trickery using Euler's formula

That was my trickery (not mine really; pretty standard trickery), not mgb_phys'. mgb_phys simply said to use the angle sum formula for sine.

a method which does not generalize to prove all other identities.

Please show one trigonometric identity that does not follow from Euler's identity, the definitions sec(x)=1/cos(x), csc(x)=1/sin(x), tan(x)=sin(x)/cos(x), cot(x)=1/tan(x), the exponential identities exp(a+b)=exp(a)*exp(b), exp(i*pi/2)=i, and exp(i*k*pi)=(-1)^k for integral k.

 

[Kurret]

Heres another way (for angles less than 90 degrees):
http://www.geocities.com/kurre999/Trigaddition.bmp

 

[tiny-tim]

Heres another way (for angles less than 90 degrees)

Hi Kurret!

Yes, that's nice too!

It only works in that form for A + B > 90º …

for A + B < 90º, your diagram becomes Georgepowell's diagram, if we make the top of your rectangle PQ, the bottom left corner T, and the right-angle of the main triangle at S.

 

[mathwonk]

just check that both satisfy the same differential equation: namely y'' + 4y = 0, and y'(0) = 2, and y(0) = 0.

 

[Georgepowell]

Heres another way (for angles less than 90 degrees):
http://www.geocities.com/kurre999/Trigaddition.bmp

Thanks, I do prefere the geometric proofs :p

EDIT: Just noticed … the angle PQO in the diagram is wrongly labelled as A (instead of 90º - A - B), … but it doesn't affect the proof.

That explains why i thought:

Doesn't it only prove it for:
2A + B = 90 ?

I incorrectly presumed it was symmetrical, and so SQO = POQ.

Another misstake in the diagram is in the top left corner i wrote "OS = cos(B)" but it doesn't, OS = cos(A).

No, I think that it works for any A + B < 90º …

Actualy i think it works for all values of A and B, as long as neither of them = 90 exactly...

http://i349.photobucket.com/albums/q390/Georgepowell77/DSC00056-1.jpg

[edit] does it still prove it for A or B = 90? - I'm not sure[edit]

 

[tiny-tim]

Another misstake in the diagram is in the top left corner i wrote "OS = cos(B)" but it doesn't, OS = cos(A).

uh … i never noticed that misprint … i was only looking at the top right corner, where you got it right anyway!

Actualy i think it works for all values of A and B, as long as neither of them = 90 exactly...

[edit] does it still prove it for A or B = 90? - I'm not sure[edit]

yes, it works for A + B > 90º too, but then your diagram turns into Kurret's

for A + B = 90º, i don't see how to adapt it … but that's just sin²A + cos²A = 1 anyway

just check that both satisfy the same differential equation: namely y'' + 4y = 0, and y'(0) = 2, and y(0) = 0.

hi mathwonk!

erm … did you wander into the wrong flat after a party? ​