This question is breaking my head (1 Viewer)

upishcat

New Member
where do I even start???

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Drdusk

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Moderator
where do I even start???
It's kind of shitty how they put literally 3 questions into one part lol. Anyway

$\bg_white \text{We have to show}\hspace{2mm}\dfrac{e^{i\alpha} + e^{i\beta}}{e^{i(\alpha + \beta)/2}} =2 \cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)$

$\bg_white LHS = \dfrac{e^{i\alpha} + e^{i\beta}}{e^{i(\alpha + \beta)/2}} = \dfrac{\cos\alpha + \cos\beta + i(\sin\alpha + \sin\beta)}{e^{i(\alpha + \beta)/2}}$

$\bg_white \text{Now remember the formulae for cos and sin}\hspace{2mm}\cos\alpha + \cos\beta = 2 \cos \bigg(\dfrac{\alpha + \beta}{2}\bigg) \cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)$

$\bg_white \therefore LHS = \dfrac{2 \cos \bigg(\dfrac{\alpha + \beta}{2}\bigg) \cos \bigg(\dfrac{\alpha - \beta}{2}\bigg) + i\Bigg(2 \sin \bigg(\dfrac{\alpha + \beta}{2}\bigg) \cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\Bigg)}{e^{i(\alpha + \beta)/2}}$

$\bg_white = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\times \dfrac{\cos \bigg(\dfrac{\alpha + \beta}{2}\bigg) + i\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)}{e^{i(\alpha + \beta)/2}}$

$\bg_white = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\times\dfrac{e^{i(\alpha + \beta)/2}}{e^{i(\alpha + \beta)/2}}$

$\bg_white \text{Which thus gives the required result.}$

upishcat

New Member
Im literally woke after seeing your explanation. Thank you so much my dude.

blizzardblizzard

New Member
woah is euler's formula in the new 4u syllabus now?

upishcat

New Member
woah is euler's formula in the new 4u syllabus now?
Yep! I personally think it a great addition. It truly is fascinating how Euler combined so many fundamental numbers in math and made a forumula out of them.

Drdusk

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Moderator
Im literally woke after seeing your explanation. Thank you so much my dude.
I just got back home so here's the rest.

$\bg_white \text{Now we know}\hspace{2mm}e^{i\alpha} + e^{i\beta} = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)e^{i(\alpha + \beta)/2}$

$\bg_white \therefore \cos\alpha + \cos\beta+ i( \sin\alpha + \sin\beta) = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)e^{i(\alpha + \beta)/2}$

$\bg_white \text{Now observe that we need to find the imaginary part of the RHS so we can equate it to the imaginary part of the LHS}$

$\bg_white \therefore RHS = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\Bigg(\cos(\alpha + \beta)/2+i\sin(\alpha + \beta)/2\Bigg)$

$\bg_white \text{Now we take the imaginary part of the RHS as we can see we need sin(alpha) + sin(beta) in the numerator}$

$\bg_white \text{Im}(RHS) =2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)$

$\bg_white \therefore \sin\alpha + \sin\beta = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)$

$\bg_white \therefore \dfrac{\sin\alpha + \sin\beta}{\cos\alpha + \cos\beta} = \dfrac{2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)}{\cos\alpha + \cos\beta}$

$\bg_white =\dfrac{2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)}{2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\cos \bigg(\dfrac{\alpha + \beta}{2}\bigg)}$

$\bg_white = \tan\bigg(\dfrac{\alpha + \beta}{2}\bigg)$

Drdusk

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Moderator
$\bg_white \text{If we want to find tan of 5pi/24 we can see if we set alpha and beta as 3pi/12 and 2pi/12 we get 5pi/24}$

$\bg_white \therefore \alpha = \pi/4, \beta = \pi/6$

$\bg_white \therefore \tan\bigg(\dfrac{5\pi}{24}\bigg) = \dfrac{2 + \sqrt{2}}{2 + \sqrt{6}}$

upishcat

New Member
I just got back home so here's the rest.

$\bg_white \text{Now we know}\hspace{2mm}e^{i\alpha} + e^{i\beta} = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)e^{i(\alpha + \beta)/2}$

$\bg_white \therefore \cos\alpha + \cos\beta+ i( \sin\alpha + \sin\beta) = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)e^{i(\alpha + \beta)/2}$

$\bg_white \text{Now observe that we need to find the imaginary part of the RHS so we can equate it to the imaginary part of the LHS}$

$\bg_white \therefore RHS = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\Bigg(\cos(\alpha + \beta)/2+i\sin(\alpha + \beta)/2\Bigg)$

$\bg_white \text{Now we take the imaginary part of the RHS as we can see we need sin(alpha) + sin(beta) in the numerator}$

$\bg_white \text{Im}(RHS) =2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)$

$\bg_white \therefore \sin\alpha + \sin\beta = 2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)$

$\bg_white \therefore \dfrac{\sin\alpha + \sin\beta}{\cos\alpha + \cos\beta} = \dfrac{2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)}{\cos\alpha + \cos\beta}$

$\bg_white =\dfrac{2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\sin \bigg(\dfrac{\alpha + \beta}{2}\bigg)}{2\cos \bigg(\dfrac{\alpha - \beta}{2}\bigg)\cos \bigg(\dfrac{\alpha + \beta}{2}\bigg)}$

$\bg_white = \tan\bigg(\dfrac{\alpha + \beta}{2}\bigg)$
But when we take the Im(RHS) shouldnt we not consider 2cos(a-b/2) because that is a real part?

Drdusk

π
Moderator
But when we take the Im(RHS) shouldnt we not consider 2cos(a-b/2) because that is a real part?
I did consider it. The Im(RHS) line in mine has the 2cos(a-b/2) in it..

upishcat

New Member
$\bg_white \text{If we want to find tan of 5pi/24 we can see if we set alpha and beta as 3pi/12 and 2pi/12 we get 5pi/24}$

$\bg_white \therefore \alpha = \pi/4, \beta = \pi/6$

$\bg_white \therefore \tan\bigg(\dfrac{5\pi}{24}\bigg) = \dfrac{2 + \sqrt{2}}{2 + \sqrt{6}}$
May i ask how you got this? I got as far as to understanding that 3pi/12 and 2pi/12 are used because we know their values using pythagoras triangles

Drdusk

π
Moderator
May i ask how you got this? I got as far as to understanding that 3pi/12 and 2pi/12 are used because we know their values using pythagoras triangles
It's from the formula we obtained for tan(alpha + beta \2). You just plug in the values of alpha and beta into the sin and cos equation.

upishcat

New Member
It's from the formula we obtained for tan(alpha + beta \2). You just plug in the values of alpha and beta into the sin and cos equation.
Ooops. How dumb of me haha.

upishcat

New Member
S
May i ask how you got this? I got as far as to understanding that 3pi/12 and 2pi/12 are used because we know their values using pythagoras triangles
shouldn't you completely rationalise the denominator to get surd form?

HeroWise

Active Member
You are a boomer if u like Eulers form

HeroWise

Active Member
I, Giorno Giovanna, hate eulers form!

Drdusk

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Moderator
I, Giorno Giovanna, hate eulers form!
Wait until uni. All the boomers will be pissed off if they even see cis. At least that's been my experience imao.

HeroWise

Active Member
interestingly I want to be a farmer when i grow up. So no uni for me amirite.
Jokes aside, aah. Interestingly, my career path atm looks really unrelated to maths rip.

stupid_girl

Active Member
I guess the question is not expecting you to apply trigonometric identities for product to sum or sum to product. Otherwise, there's no point to bring complex number into the question. Having said that, I think a trigonometric approach is more elegant, especially for the second part of the question.

$\bg_white \frac{e^{i\alpha}+e^{i\beta}}{e^{i\frac{\alpha+\beta}{2}}}=\frac{e^{i\alpha}}{e^{i\frac{\alpha+\beta}{2}}}+\frac{e^{i\beta}}{e^{i\frac{\alpha+\beta}{2}}}=e^{i\frac{\alpha-\beta}{2}}+e^{i\frac{\beta-\alpha}{2}}=\cos\frac{\alpha-\beta}{2}+i\sin\frac{\alpha-\beta}{2}+\cos\frac{\beta-\alpha}{2}+i\sin\frac{\beta-\alpha}{2}=2\cos\frac{\alpha-\beta}{2}$

$\bg_white e^{i\alpha}+e^{i\beta}=2\cos\frac{\alpha-\beta}{2}e^{i\frac{\alpha+\beta}{2}}$

$\bg_white \frac{\sin\alpha+\sin\beta}{\cos\alpha+\cos\beta}=\frac{Im\left(e^{i\alpha}+e^{i\beta}\right)}{Re\left(e^{i\alpha}+e^{i\beta}\right)}=\frac{2\cos\frac{\alpha-\beta}{2}\sin\frac{\alpha+\beta}{2}}{2\cos\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2}}=\tan\frac{\alpha+\beta}{2}$

upishcat

New Member
I guess the question is not expecting you to apply trigonometric identities for product to sum or sum to product. Otherwise, there's no point to bring complex number into the question. Having said that, I think a trigonometric approach is more elegant, especially for the second part of the question.

$\bg_white \frac{e^{i\alpha}+e^{i\beta}}{e^{i\frac{\alpha+\beta}{2}}}=\frac{e^{i\alpha}}{e^{i\frac{\alpha+\beta}{2}}}+\frac{e^{i\beta}}{e^{i\frac{\alpha+\beta}{2}}}=e^{i\frac{\alpha-\beta}{2}}+e^{i\frac{\beta-\alpha}{2}}=\cos\frac{\alpha-\beta}{2}+i\sin\frac{\alpha-\beta}{2}+\cos\frac{\beta-\alpha}{2}+i\sin\frac{\beta-\alpha}{2}=2\cos\frac{\alpha-\beta}{2}$

$\bg_white e^{i\alpha}+e^{i\beta}=2\cos\frac{\alpha-\beta}{2}e^{i\frac{\alpha+\beta}{2}}$

$\bg_white \frac{\sin\alpha+\sin\beta}{\cos\alpha+\cos\beta}=\frac{Im\left(e^{i\alpha}+e^{i\beta}\right)}{Re\left(e^{i\alpha}+e^{i\beta}\right)}=\frac{2\cos\frac{\alpha-\beta}{2}\sin\frac{\alpha+\beta}{2}}{2\cos\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2}}=\tan\frac{\alpha+\beta}{2}$
Sorry DrDusk but I think I will have to go with this one. Stupid_girl's got a point.