# New PDF release: A Course in Real Analysis

By J. McDonald, N. Weiss

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Donn´e 0 ≤ x ≤ π, il existe donc un et un seul nombre −1 ≤ y ≤ 1 tel que arccos y = x. Les fonctions trigonom´etriques cosinus et sinus sont d´efinies pour 0 ≤ x ≤ π par les relations cos x = y , sin x = 1 − y 2 . Elles sont prolong´ees `a l’axe r´eel R tout entier en posant d’abord, pour −π < x < 0, cos x = cos(−x) , sin x = − sin(−x) et ensuite, pour n ∈ Z, cos(x + 2πn) = cos x , sin(x + 2πn) = sin x. Observons les valeurs remarquables π 1 π 3π 1 = √ , cos = 0 , cos = − √ , cos π = −1 4 2 4 2 2 cos 0 = 1 , cos et π 1 π 3π 1 = √ , sin = 1 , sin = √ , sin π = 0.

En voici un exemple. Soit N ≥ 2 un entier naturel. Alors sinN x dx = = = sinN −2 x dx − sinN −2 x dx − = sinN −2 x dx − (sinN −2 x cos x) cos x dx sinN −1 x cos x + N −1 sinN −2 x dx − sinN −2 x cos2 x dx sinN −1 x sin x dx N −1 sinN −1 x cos x − N −1 51 sinN x dx N −1 de telle sorte que sinN x dx 1 + 1 N −1 sinN −2 x dx − = sinN −1 x cos x . N −1 Autrement dit : sinN x dx = − sinN −1 x cos x N − 1 + N N sinN −2 x dx. (10) La formule de Wallis est une belle application de cette derni`ere relation.

2 Si m = n, on en tire +π cos mx cos nx dx = −π sin(m − n)x sin(m + n)x + m−n m+n 1 2 π −π =0 alors que si m = n, on obtient +π cos2 mx dx = −π 1 2 x+ sin 2mx 2m π −π = π. Les autres cas sont similaires. D. 3 Les fonctions trigonom´ etriques inverses La fonction arccosinus (figure (12)), arccos : [−1, 1] → [0, π], est d´efinie par la relation 1 1 − t2 dt + y arccos y = 2 1 − y2 y comme nous l’avons vu. Elle est continue sur [−1, 1] et d´erivable sur ]−1, 1[, avec d −1 . arccos y = dy 1 − y2 41 C’est une fonction strictement d´ecroissante et l’on a cos(arccos y) = y , y ∈ [−1, 1] et arccos(cos x) = x , x ∈ [0, π].

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### A Course in Real Analysis by J. McDonald, N. Weiss

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