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Given that cos a + cos b + cos g = 0 and sin a + sin b + sin g = 0, which of the following identities holds?

1. cos2a + cos2b + cos2g = sin2a + sin2b + sin2g = 0
2. sin3a + sin3b + sin3g = 3*sin(a + b + g)
3. cos3a + cos3b + cos3g = 3*cos(a + b + g)
4. all of these

Correct answer: all of these

Solution

All three standard results follow from p + q + r = 0: the sum of squares gives the cos2/sin2 result, and p³ + q³ + r³ = 3pqr gives the cos3 and sin3 results.

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