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dresu9dnjn

dresu9dnjn

Answered question

2022-05-12

To prove:
sin ( x ) + sin ( 3 x ) = 2 cos ( x ) sin ( 2 x )
The trigonometric rule sin ( x + y ) = sin ( x ) cos ( y ) + cos ( x ) sin ( y ) didn't get me that far

Answer & Explanation

kazneni3tr2b

kazneni3tr2b

Beginner2022-05-13Added 17 answers

Use :
sin A + sin B = 2 sin ( A + B 2 ) cos ( A B 2 )
Therefore :
sin ( 3 x ) + sin ( x ) = 2 sin ( 3 x + x 2 ) cos ( 3 x x 2 ) = 2 cos ( x ) sin ( 2 x )
Tristan Meyers

Tristan Meyers

Beginner2022-05-14Added 5 answers

Note the following: (1) sin ( A + B ) = sin A cos B + sin B cos A
Also note that (2) sin ( A B ) = sin A cos B sin B cos A
( 1 ) + ( 2 ) = sin ( A + B ) + sin ( A B ) = 2 sin A cos B
Then let A+B=x,A-B=3x
We then get A = 3 x + B 3 x + 2 B = x B = x A = 2 x
Then substituting this in we arrive at
sin x + sin 3 x = 2 sin ( 2 x ) cos ( x ) = 2 sin ( 2 x ) cos ( x ) as required

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