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Câu 37 trang 121 SGK Đại số và Giải tích 11 Nâng cao

Bốn góc lượng giác
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Đề bài

ꦯSố đo bốn góc của một tứ giác lồi lập thành một cấp số nhân. Hãy tìm bốn góc đó, biết rằng số đo của góc lớn nhất gấp 8 lần số đo của góc nhỏ nhất.

Phương pháp giải - Xem chi tiết

- Sử dụng tổng số đo các góc của một tứ giác bằng \(360^0\) - Công thức số hạng tổng quát tìm q:\[{u_n} = {u_1}{q^{n - 1}}\] - Công thức tổng n số hạng đầu tìm số đo góc nhỏ nhất: \[{S_n} = \frac{{{u_1}\left( {1 - {q^n}} \right)}}{{1 - q}}\]

Lời giải chi tiết

Kí hiệu A, B, C, D là số đo bốn góc (tính theo đơn vị độ) của tứ giác lồi đã cho.Không mất tổng quát, giả sử \(A ≤ B ≤ C ≤ D\).Khi đó, từ giả thiết của bài toán ta có \(D = 8A\), và A, B, C, D theo thứ tự đó lập thành một cấp số nhân.Gọi q là công bội của cấp số nhân đó, ta có :\(8A = D = A.q^3\)\( \Leftrightarrow {q^3} = 8\)\(⇔ q = 2\).Do đó \(360 ^0= A + B + C + D \)\(= A.{{1 - {2^4}} \over {1 - 2}} = 15A \Leftrightarrow A = 24^0\)Suy ra \(B = A.2 = 48^0\), \(C = A.2^2= 96^0\) và \(D = A.2^3= 192\)

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