Proving sin &#x2061;<!-- ⁡ --> ( 120 <mrow

Brennen Fisher

Brennen Fisher

Answered question

2022-05-31

Proving sin ( 120 ) 1 + sin ( 90 + a ) sin ( 240 ) 1 cos ( a ) = 3 sin 2 ( a )

Answer & Explanation

rass1k6s

rass1k6s

Beginner2022-06-01Added 13 answers

We have that sin ( 90 + a ) = cos ( a ) = cos ( a ). Moreover
sin 2 ( a ) = 1 cos 2 ( a ) = ( 1 + cos ( a ) ) ( 1 cos ( a ) ) .
Therefore
sin ( 120 ) 1 + sin ( 90 + a ) sin ( 240 ) 1 cos ( a ) = sin ( 120 ) ( 1 cos ( a ) ) sin ( 240 ) ( 1 + cos ( a ) ) sin 2 ( a ) = 3 sin 2 ( a )
where in the last step we used sin ( 120 ) = sin ( 240 ) = 3 / 2
Akira Huang

Akira Huang

Beginner2022-06-02Added 3 answers

Use the relations: sin ( 90 + a ) = cos a, cos a = cos ( a ) and sin ( 120 ) = sin ( 240 ). Then the LHS of identity become
sin ( 120 ) 1 + cos a + sin ( 120 ) 1 cos a = 2 sin ( 120 ) sin 2 a = 2 cos ( 30 ) sin 2 a = 2 3 2 sin 2 a = 3 sin 2 a .

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