If #sin alpha + sin beta + sin gamma = cos alpha + cos beta + cos gamma# then what is the value of #cos^2 alpha + cos^2 beta + cos^2 gamma# ?

2 Answers
George C.
May 27, 2016

There is insufficient information to determine a single value.

Explanation:

The question gives insufficient information to determine a unique value.

For example:

If #(alpha, beta, gamma) = (0, pi, pi/4)# then:

#{ (sin alpha + sin beta + sin gamma = 0+0+sqrt(2)/2 = sqrt(2)/2), (cos alpha + cos beta + cos gamma = 1-1+sqrt(2)/2 = sqrt(2)/2), (cos^2 alpha+cos^2beta+cos^2gamma = 1+1+1/2 = 5/2) :}#

If #(alpha, beta, gamma) = (pi/4, pi/4, pi/4)# then:

#{ (sin alpha + sin beta + sin gamma = sqrt(2)/2+sqrt(2)/2+sqrt(2)/2 = (3sqrt(2))/2), (cos alpha + cos beta + cos gamma = sqrt(2)/2+sqrt(2)/2+sqrt(2)/2 = (3sqrt(2))/2), (cos^2 alpha+cos^2 beta + cos^2 gamma = 1/2+1/2+1/2 = 3/2) :}#

Are #alpha, beta, gamma# supposed to be the internal angles of a triangle or something like that? Is there some information missing from the question?

bp
May 31, 2016

Subject to #alpha +beta +gamma =pi#\
#cos^2 alpha +cos^2 beta +cos^2 gamma= 3/2 +cos alpha +cos beta +cos gamma = 3/2 +sin alpha +sin beta +sin gamma#

Explanation:

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also #cos^2 alpha +cos^2 beta+ cos^3 gamma = 3/2 + sin alpha +sin beta + sin gamma #

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