If &#x03B1;<!-- α --> , &#x03B2;<!-- β --> , &#x03B3;<!-- γ --> be three distinct

Scolfaro2y

Scolfaro2y

Answered question

2022-05-18

If α , β , γ be three distinct real values such that
sin α + sin β + sin γ sin ( α + β + γ ) = cos α + cos β + cos γ cos ( α + β + γ ) = 2
Then value of cos ( α + β ) + cos ( β + γ ) + cos ( γ + α ) =

Answer & Explanation

Sasha Pacheco

Sasha Pacheco

Beginner2022-05-19Added 10 answers

Let   p p i = ( cos   θ i ,   sin   θ i )   and   p p = ( cos i θ i ,   sin i θ i )   ( i = 1 , 2 , 3 ) .. Then i p p i = 2 p p ,, and so ( i   p p i ) p p = 2.
zato1kom7

zato1kom7

Beginner2022-05-20Added 2 answers

Write sin ( α ) as sin ( t ( β + γ ) ) where t = α + β + γ Now, expand.
We get
cos ( β + γ ) + cos ( γ + α ) + c o s ( α + β ) cot t ( sin ( β + γ ) + sin ( γ + α ) + s i n ( α + β ) ) = cos ( β + γ ) + cos ( γ + α ) + c o s ( α + β ) + tan t ( sin ( β + γ ) + sin ( γ + α ) + s i n ( α + β ) ) = 2
Now, tant can't be equal to cot t
So, sin ( β + γ ) + sin ( γ + α ) + s i n ( α + β ) = 0
Substituting back in the above equation,
cos ( β + γ ) + cos ( γ + α ) + c o s ( α + β ) = 2

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