How to simplify y = sin &#x2061;<!-- ⁡ --> ( arcsin &#x2061;

Ayanna Trujillo

Ayanna Trujillo

Answered question

2022-06-13

How to simplify y = sin ( arcsin ( x ) n ) , n 1?

Answer & Explanation

Brendon Fernandez

Brendon Fernandez

Beginner2022-06-14Added 14 answers

Let θ = arcsin x and we have to assume that | x | < 1 2
Then tan θ = x 1 x 2 and | tan θ | < 1
cos θ n + i sin θ n = ( cos θ + i sin θ ) 1 n = ( cos θ ) 1 n ( 1 + i tan θ ) 1 n = x 1 n ( 1 + i x 1 x 2 ) 1 n = x 1 n k = 0 ( 1 / n k ) i k ( x 1 x 2 ) k cos θ n = x 1 n k = 0 ( 1 ) k ( 1 / n 2 k ) ( x 2 1 x 2 ) k sin θ n = x 1 n x 1 x 2 k = 0 ( 1 ) k ( 1 / n 2 k + 1 ) ( x 2 1 x 2 ) k

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