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Trigo-no-metry

If

\(cos \alpha + cos \beta +cos \gamma = sin \alpha + sin \beta +sin \gamma = 0 \)

Then Prove that

\(cos2 \alpha + cos2 \beta +cos2 \gamma = sin2 \alpha + sin2 \beta +sin2 \gamma = 0 \)

Note by Sudipta Biswas
2 years, 10 months ago

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Hi there! Besides complex numbers and/or vectors, I guess a purely trigo proof would be interesting as well (don't worry, it actually isn't at all that messy either...)

So we are given that \( \sin \gamma = - \sin \alpha - \sin \beta \) and \( \cos \gamma = - \cos \alpha - \cos \beta \). Then, using the identity \( \sin^2 \gamma + \cos^2 \gamma = 1\), and expanding, this results in \( 2 + 2 \sin \alpha \sin \beta + 2 \cos \alpha \cos \beta = 1 \), and simplifying, we may obtain that \( \cos ( \alpha - \beta ) = - \frac{1}{2} \), or in other words, the difference between \( \alpha \) and \( \beta \) is 120 degrees. Similarly, since the angles are symmetric, we can say that the difference between \( \beta \) and \( \gamma \), as well as the difference between \( \gamma \) and \( \alpha \), are 120 degrees as well. In other words, when drawn as the angle from the positive x-axis on cartesian axes, these represent 3 equally spaced angles through the 'cycle' of 360 degrees.

Now, we may then WLOG assume \( \beta = \alpha + 120 ^ \circ \) and \( \gamma = \alpha - 120 ^ \circ \).

Hence, \( \sin 2 \alpha + \sin 2 \beta + \sin 2 \gamma \) \( = \sin 2 \alpha + \sin ( 2 \alpha + 240 ^ \circ ) + \sin ( 2 \alpha - 240 ^ \circ ) \) \( = \sin 2 \alpha + 2 \sin 2 \alpha \cos 240 ^ \circ \) \( = \sin 2 \alpha - \sin 2 \alpha = 0 \)

Similarly, \( \cos 2 \alpha + \cos 2 \beta + \cos 2 \gamma \) \( = \cos 2 \alpha + \cos ( 2 \alpha + 240 ^ \circ ) + \cos ( 2 \alpha - 240 ^ \circ ) \) \( = \cos 2 \alpha + 2 \cos 2 \alpha \cos 240 ^ \circ \) \( = \cos 2 \alpha - \cos 2 \alpha = 0 \) Jau Tung Chan · 2 years, 10 months ago

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This problem has a familiar ring to it, doesn't it? Something about complex numbers (or maybe vectors). But I don't want to spoil it with the solution. Michael Mendrin · 2 years, 10 months ago

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@Michael Mendrin One can also prove that-

\( cos3α + cos3β + cos3ϒ = 3 ~cos(α + β + ϒ) \)

\( sin3α + sin3β + sin3ϒ = 3 ~sin(α + β + ϒ) \) Avineil Jain · 2 years, 10 months ago

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@Michael Mendrin Right, we want to show that if \( |a|=|b|=|c| = 1 \) and \( a+b+c = 0 \) then \( a^2+b^2+c^2 = 0 \). Pretty straightforward. There is also a nice connection with equilateral triangles... Patrick Corn · 2 years, 10 months ago

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it can be prooved by usng complex numbers Dharma Teja · 2 years, 10 months ago

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why y=a-120 Yee Cheng · 2 years, 9 months ago

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How do you answer this? Kent Mercado · 2 years, 10 months ago

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Ah very nice. I set this question in the practice section. It's an interesting result. Calvin Lin Staff · 2 years, 10 months ago

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