Cho tam giác ABC. Chứng minh sinA/2 . sinB/2 . sinC/2 ≤ 1/8

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Cho tam giác ABC. Chứng minh sinA/2 . sinB/2 . sinC/2 ≤ 1/8

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Valerie 4 tháng 2021-08-25T07:30:50+00:00 1 Answers 147 views 0

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    2021-08-25T07:32:07+00:00

    `=>` Tặng bạn 

    Đặt   `P = sinA/2.sinB/2.sinC/2`.

    `<=> 2P = ((2sinA)/2*(2.sinB)/2).(2sinC)/2 `

    `<=> 2P = [cos(A/2-B/2) – cos(A/2+B/2)].sin(C/2) `

    `<=> 2P = [cos(A/2-B/2) – sin(C/2)].sin(C/2)` `<=> 2P= sin(C/2).cos(A/2-B/2) – sin²(C/2)` 

    `<=> 8P = 4sin(C/2).cos(A/2-B/2) – 4sin²(C/2)` 

    `<=> 1-8P = 4sin²(C/2) – 4sin(C/2).cos(A/2-B/2) + cos²(A/2-B/2) + 1 – cos²(A/2-B/2) `

    `<=> 1-8P = [2sin(C/2) – cos(A/2-B/2)]² + sin²(A/2-B/2) ≥ 0 (*) `

    `=> P ≤ 1/8`

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