Justifying sin &#x2061;<!-- ⁡ --> ( ( n +

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Answered question

2022-06-22

Justifying sin ( ( n + 1 2 ) π ) = ( 1 ) n without a calculator

Answer & Explanation

Daniel Valdez

Daniel Valdez

Beginner2022-06-23Added 19 answers

By induction.
For n=0 one has sin π 2 = 1
Assume that the identity is valid for n−1.
Let's write
sin ( n + 1 2 ) π = sin ( π + ( n 1 + 1 2 ) π ) = sin ( π + α )
with α = ( n 1 + 1 2 ) π. Now we know that α , sin ( π + α ) = sin α and therefore
sin ( n + 1 2 ) π = sin ( n 1 + 1 2 ) π = ( 1 ) n 1 = ( 1 ) n

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