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Giải mục 2 trang 62, 63 SGK Toán 10 tập 1 - Chân trời sáng tạo

Trên nửa đường tròn đơn vị, cho dây cung NM song song với trục Ox (Hình 4). Tính tổng số đo của hai góc xOM và xON Tính các giá trị lượng giác: sin120, cos150;cot 135

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HĐ Khám phá 2

Trên nửa đường tròn đơn vị, cho dây cung NM song song với trục Ox (Hình 4). Tính tổng số đo của hai góc \(\widehat {xOM}\) và \(\widehat {xON}.\)

Phương pháp giải:

Tính góc \(\widehat {xON}\) theo góc \(\widehat {xOM}.\)

Lời giải chi tiết:

Gọi H là hình chiếu vuông góc của N Ox.

 

Ta có: \(\widehat {NOH} = \widehat {ONM} = \widehat {OMN} = \widehat {MOx} = \alpha \) (do NM song song với Ox) Mà \(\widehat {xOM} + \widehat {NOH} = {180^o}\) Suy ra \(\widehat {xON} + \widehat {MOx} = {180^o}\)

Thực hành 2

Tính các giá trị lượng giác: \(\sin {120^o};\cos {150^o};\cot {135^o}.\)

Phương pháp giải:

\(\begin{array}{l}\sin {120^o} = \sin \;({180^o} - {60^o});\\\cos {150^o} =  - \cos \;({180^o} - {30^o});\\\cot {135^o} =  - \cot \;({180^o} - {45^o}).\end{array}\)

Lời giải chi tiết:

\(\begin{array}{l}\sin {120^o} = \sin \;({180^o} - {60^o}) = \sin {60^o} = \frac{{\sqrt 3 }}{2};\\\cos {150^o} =  - \cos \;({180^o} - {30^o}) =  - \cos {30^o} =  - \frac{{\sqrt 3 }}{2};\\\cot {135^o} =  - \cot \;({180^o} - {45^o}) =  - \cot {45^o} =  - 1.\end{array}\)

Vận dụng 1

Cho biết \(\sin \alpha  = \frac{1}{2},\) tìm góc \(\alpha \;({0^o} \le \alpha  \le {180^o})\) bằng cách vẽ nửa đường tròn đơn vị.

Phương pháp giải:

Vẽ nửa đường tròn đơn vị. \(\sin \alpha  = \frac{1}{2}\) nên lấy các điểm có tung độ là \(\frac{1}{2}\). Từ đó tính góc \(\alpha \).

Lời giải chi tiết:

Gọi M là điểm thuộc nửa đường tròn đơn vị sao cho: \(\widehat {xOM} = \alpha \) Do \(\sin \alpha  = \frac{1}{2}\) nên tung độ của M bằng \(\frac{1}{2}.\) Vậy ta xác định được hai điểm N và M thỏa mãn \(\sin \widehat {xON} = \sin \widehat {xOM} = \frac{1}{2}\)

Đặt \(\beta  = \widehat {xOM} \Rightarrow \widehat {xON} = {180^o} - \beta \) Xét tam giác OHM vuông tại H ta có: \(MH = \frac{1}{2} = \frac{{OM}}{2} \Rightarrow \beta  = {30^o}\) \( \Rightarrow \widehat {xON} = {180^o} - {30^o} = {150^o}\) Vậy \(\alpha  = {30^o}\) hoặc \(\alpha  = {150^o}\)

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