Proof of sin &#x2061;<!-- ⁡ --> ( 3 &#x03C0;<!-- π --> </mrow

Hayley Sanders

Hayley Sanders

Answered question

2022-05-28

Proof of sin ( 3 π 2 A ) = cos ( A ) geometrically

Answer & Explanation

relientaaho2

relientaaho2

Beginner2022-05-29Added 13 answers

Assume | A | < π, and let a = ( u , v ) S 1 represent the angle A. Then a ¯ = ( u , v ) represents the angle −A. In order to obtain the point b representing angle B := A + 3 π 2 we have to rotate a ¯ by 3 π 2 counterclockwise, or what is the same thing: by π 2 clockwise. Such a rotation has the effect ( x , y ) ( y , x ) on points (x,y). It follows that b = ( v , u ), so that we obtain
sin ( 3 π 2 A ) = sin B = u = cos A   , ,
as claimed

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