Considere a circunferência de raio ![[;\overline{OA};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vn_JNUxde9SaTEnPmi5hlqH_Wx0hep594U9fdb35uoGxvKF9qut-gl0p3PAx-Q3O4JAKep6SrG8F7s2JlWfB7a-Hmpe64wES8aONQ=s0-d "\overline{OA}")
centrada na origem ilustrada abaixo e tome um ponto P qualquer no primeiro quadrante da mesma tal que ![[;\angle{POA}=x;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vtU0GnX5SUsBNjGn1BBvulhalN2h6ByC9Vi-O2z5BCYJElO_aiArCRO5v2fOKjp1cRL-8wU-PQhgNrqSYHSoi85v20kzOYEIE--g=s0-d "\angle{POA}=x")
.
centrada na origem ilustrada abaixo e tome um ponto P qualquer no primeiro quadrante da mesma tal que .
Traçando a circunferência de raio ![[;\overline{PM};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sGU6zBmO5V_uTv3doJ3YZdFGvRe5nZQqBsn-sWx6DP40M2MXGRsByeaZF33PR-Wp5FVekHQpGsW_7Xb2sOt0o3_CsV9j-KohNjgA=s0-d "\overline{PM}")
![[;\perp \overline{OA};]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vEVyZGiaHCxy1NjoimvDaz_YO87-iPifyKxGPI-MAwZB6xGMHlsW6lkQEZOo99UJtJDaN71bgauzZIIuHMS2TwMdKDzN8-FmIxldDEf5b0vAFBIUor=s0-d "\perp \overline{OA}")
, temos que:
, temos que:![[;m(PQ) = 2\sin(x);]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tmNLNSsauZfl4pnH3sxYxSLjjsd3x2j24mx6yqjr7D5GI0mtJDmyuZUYRulLdxmJCy8s3Yi6-2ULx8TyPQVz5iwgqA_YtLnpduv-Ky84_kZYPztYxN-A=s0-d "m(PQ) = 2\sin(x)")
![[;m(\widehat{PAQ}) = 2x;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sKHuEQrtvl8boun7KasRCS6qVsv179OCxlvRfpmTRRSGdiakhwRqgH_E_muKfV3L3eEuR4pwZdhiwwa0cAX6QRZA_Hj08PsDyXscBIKnEo-4xB4XURu-TpUaGy=s0-d "m(\widehat{PAQ}) = 2x")
![[;m(\widehat{PBQ}) = \pi \sin(x);]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sY962i9Qe8HRsmjZnDrO3r-UGlO_S1k6sOt8IMFux9Egq13KZnkKEyqNfHohqSlaUngUissn46RLc8AcdIs84GnlPmgECWL3mhrQK5GL4NKFrNBVfuRIU0rAf4JlcTYNlcpAgXFp3WyWJNovTl=s0-d "m(\widehat{PBQ}) = \pi \sin(x)")
Mas:
![[;m(PQ) \leq m(\widehat{PAQ}) \leq m(\widehat{PBQ});]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t9GnU3su5moWywi-53zusyFWYN0fWqrQoED6VndWScJN8Y8svN8067nnEpjQFORk4q_h6s6pvttrH9WzDVxDEqxKB8SxfQnpdKzJxu9pcGRrwi7C94e9mAPQRwjeXyzUkHxzpYsxGmZX5V1g-cBTYJRymU0QUsSg6tjLgx8AWuE_vshRioVbhhjh32uWQ=s0-d "m(PQ) \leq m(\widehat{PAQ}) \leq m(\widehat{PBQ})")
Da primeira parte da desigualdade vem ![[;\sin(x)\leq x;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_smgGYIsuVYX1Li91Ypnethrsw1dULbdoojL2ePzZjUDBIJ8AGEiNfY9uGmo2I0sMecvEVqOqveEvj5S9hBSFrvMToLNESUk8oSTguZpA8e=s0-d "\sin(x)\leq x")
e da segunda vem ![[;\frac{2}{\pi}x \leq \sin(x);]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t-CBwe5aGSEmiH1ZQ_JzZjKUxSCDKUOTdLGEXy4xgQ8MeQlb_yxXakpdGS2W_mw7LWcKmjltVw445cGQaSmN96g3x7MjLUHKP7d29odBpeuXCvebkzDfwf4AXBTSxmdp3ODo386y0QeScW=s0-d "\frac{2}{\pi}x \leq \sin(x)")
. Logo:
e da segunda vem . Logo:![[;\leq x;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vn-OUCQEcmL7-rdwWp6nNuzTRkHk2sehwUH7hFAi9oKK2cb2Zwk561DhsiU6O_SO6CEE1Q65-4t1vapOzLGuAfdYs=s0-d "\leq x")
[CQD]
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