Showing that sin &#x2061;<!-- ⁡ --> ( &#x03C0;<!-- π --> 2 </mfrac>

Mara Cook

Mara Cook

Answered question

2022-06-13

Showing that sin ( π 2 x ) = cos x using complex exponentials
This is where I've got to:
sin ( π 2 x ) = i 2 ( e x i π i 2 e π 2 x i )
What should I do next?

Answer & Explanation

Cahokiavv

Cahokiavv

Beginner2022-06-14Added 31 answers

sin ( π 2 x ) = 1 2 i ( e i ( π 2 x ) e i ( π 2 x ) ) = 1 2 i ( e i π 2 i x e i π 2 + i x ) = 1 2 i ( e i π 2 ( e i x ) e i π 2 ( e i x ) ) = 1 2 i ( i ( e i x ) + i ( e i x ) ) = 1 2 ( e i x + e i x ) = 1 2 ( e i x + e i x ) = cos ( x )
Where I have used Euler's formula to find
e i π 2 = cos ( π 2 ) + i sin ( π 2 ) = i
e i π 2 = cos ( π 2 ) + i sin ( π 2 ) = i

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?