ᠬᠤᠷᠢᠨ ᠨᠠᠢᠮᠠᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠬᠤᠷᠴᠠ ᠦᠨᠴᠦᠭ ᠤᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠹᠦᠩᠺᠼ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠬᠣᠷ ᠢᠨ ᠨᠠᠢ ᠮᠠᠳᠤ ᠭᠠᠷ ᠪᠥᠯᠥ ᠭ ᠦᠨ ᠨᠢᠭᠡ ᠴᠢ ᠶᠢ ᠨ ᠪᠠᠢᠴᠠ ᠭᠠᠯᠲᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ (90 ᠮᠢᠨᠦ᠋ ᠲ᠂ ᠨᠠᠢᠲᠡ 12 0 ᠬ ᠤᠪᠢ) ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰ ᠡᠲᠤᠪ᠃ (ᠰᠡ ᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤ ᠪᠢ) 1. ᠬᠡᠷᠪᠡ ᠥᠨᠴ ᠥᠭ 𝛼 ᠨᠢ ᠬᠤᠷᠴᠠ ᠥᠨ ᠴᠥᠭ ᠮᠥ ᠷᠳᠡᠭᠡ ᠨ𝑠𝑖𝑛𝛼 = 45 ᠪᠣᠯ᠂ 𝑡𝑎𝑛𝛼 = ︵ ︶ A. 925 B. 35 C. 34 D. 43 2. Rt ∆AB C ᠳᠦ᠂ ∠C = 90 ° ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠡᠴᠡ ᠯᠠᠪᠲᠠ ᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠭᠳᠠᠬᠤ ᠨᠢ ( ) A. 𝑠𝑖𝑛𝐴 = 𝑠𝑖𝑛𝐵 B. 𝑐𝑜𝑠𝐴 = 𝑠𝑖𝑛𝐵 C. 𝑠𝑖𝑛𝐴 = 𝑐𝑜𝑠𝐵 D. ∠A + ∠B = 90° 3. ᠲᠡ ᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠬᠣᠶ ᠠᠷ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 6 , 8᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠭᠤ ᠷᠪᠠᠳᠤᠭᠠ ᠷ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ ( ) A. 10 B. 2 √2 C. 10 ᠪᠤᠶ ᠤ 2 √7 D . ᠳᠤᠭᠲᠠᠭ ᠠᠵᠤ ᠴᠢᠳᠠ ᠬᠤ ᠦ ᠭᠡᠢ 4. Rt ∆AB C ᠳᠦ᠂ ∠C = 90 °᠂ ᠬᠡᠷᠪᠡ ᠦᠭ ᠭᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠨᠢ ∠A ᠪᠠ 𝑎 ᠦᠶ ᠡᠰ᠂ 𝑐 ᠶᠢ ᠣ ᠯᠪᠠᠯ᠂ ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠬᠣᠯ ᠪᠣᠭᠳᠠᠯ ᠢᠯ ᠡᠷᠭᠡᠢᠯ ᠡᠯ ( ) ᠶᠢ ᠰᠤ ᠩᠭᠤᠨ ᠠ᠃ A. 𝑐= 𝑎𝑠𝑖𝑛𝐴 B. 𝑐= 𝑎𝑐𝑜𝑠𝐴 C. 𝑐= 𝑎∙ 𝑡𝑎𝑛𝐴 D. 𝑐= 𝑎𝑡𝑎𝑛𝐴 5. 𝑠𝑖𝑛 45° + 𝑐𝑜𝑠 45° =( ) A. √2 B. √3 :12 C. √3 D. 1 6. Rt ∆AB C ᠳᠦ᠂ ∠C = 90 ° 𝑡𝑎𝑛𝐴 = 3᠂ 𝐴𝐶 = 10 ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ 𝑆∆ 𝐴𝐵𝐶 =( ) A. 3 B. 300 C. 503 D . 15 7. ᠬᠤᠷᠴᠠ ᠥᠨᠴ ᠥᠭ α 30° ᠦ ᠶᠡᠰ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ co sα ᠦᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ ( ) A. 12 ᠡᠴᠡ ᠶᠡᠭᠡ B. 12 ᠡᠴᠡ ᠪᠠᠭ ᠠ C. √32 ᠡᠴᠡ ᠶᠡᠭᠡ D. √32 ᠡᠴᠡ ᠪᠠᠭ ᠠ 8. ᠪᠠᠲᠤ ᠨᠠ ᠯᠠᠤ ᠶᠢᠨ ᠥᠨᠴᠥ ᠭ ᠨᠢ 30 ° ᠪᠣ ᠯᠬᠤ ᠨᠠ ᠯᠠᠤ ᠳᠠᠭ ᠠᠤ 2 𝑚 ᠶᠠᠪᠤᠭ ᠰᠠᠨ ᠪᠣᠯ᠂ ᠲᠥᠪᠰᠢ ᠨ ᠨᠢ ( ) m ᠳᠣᠷ ᠣᠭᠰ ᠢᠯᠠᠨ ᠠ᠃ A. 1 B . √ 3 C. 2√ 3 D . 2 √33 9. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ ABCD ᠳᠦ᠂ ∠A =6 0°᠂ ∠B= ∠D =9 0°᠂ BC= 2᠂ CD =3 ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ A B= ( ) A. 4 B. 5 C. 2√ 3 D. 8√ 33 10 . ᠦᠭ ᠭᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠨᠢ Rt ∆AB C ᠳᠦ᠂ ∠C = 90° ᠂ 𝑡𝑎𝑛𝐴 = 43 ᠂ 𝐵𝐶 = 8᠂ ᠲᠡᠭᠡᠪ ᠡᠯ AC =( ) A. 6 B. 323 C . 1 0 D . 12 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠣ ᠭᠣᠰ ᠤᠨ ᠳᠥ ᠲᠠᠭᠯᠠ ᠬᠤ ᠰᠡ ᠳᠦᠪ ᠃ ( ᠰ ᠡᠳᠤᠪ ᠪᠦᠷᠢ 4 ᠬᠤᠪ ᠢ᠂ ᠨᠡ ᠢᠲᠠ 28 ᠬᠤᠪᠢ ) 11 . ᠪᠣ ᠳᠣ᠃ 2s in3 0° +2 co s60 °+ 3ta n4 5° =_ __ __ __ 12 . ᠬᠡ ᠷᠪᠡ sin 28 °=c osα ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ α=_ __ __ __ _ 13 . ᠦ ᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ∆𝐴𝐵𝐶 ᠳᠦ᠂ ∠C=9 0°᠂ AB =1 3᠂ AC=5 ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ tan A= __ __ __ 14 . ᠨᠢ ᠭᠡᠨ ᠨᠠᠯᠠᠤ ᠶᠢᠨ ᠨᠠᠯᠠᠤ ᠴᠠ ᠨᠢ 1 :√ 3᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠨᠠ ᠯᠠᠤ ᠶᠢᠨ ᠥᠨᠴᠥ ᠭ ᠨᠢ __ __ __ ᠭᠷᠠᠳᠦ᠋ᠰ 15 . Rt ∆AB C ᠳᠦ᠂ ∠C = 90 °᠂ 𝐴𝐵 ＝10 𝑐𝑚 ᠂ 𝑠𝑖𝑛𝐴 ＝ 45 ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ 𝐵𝐶 ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ_ __ __ __ cm 16. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠨᠢᠭᠡ ᠦᠨᠳᠤᠷ ᠠᠰᠠᠷ ᠤᠨ ᠡᠮᠤᠨ ᠡ D ᠴᠡᠭ ᠡᠴᠡ ᠠᠰᠠᠷ ᠤᠨ ᠣ ᠷᠣᠢ ᠶᠢ ᠨ ᠴᠡ ᠭ ᠦᠨ ᠥᠯᠢᠢ ᠭᠰᠡᠨ ᠥᠨᠴ ᠥᠭ ᠢ ᠬᠡᠮᠵᠢ ᠭᠰᠡᠨ ᠳᠦ 30° ᠂ ᠡᠮᠤᠨᠡᠰᠢ 60 m ᠶᠠᠪᠤᠭ ᠠᠳ 𝐶 ᠴᠡ ᠭ ᠲᠦ ᠣᠴᠢᠭᠰ ᠠᠨ ᠤ ᠳᠠᠷᠠᠭ ᠠ ᠲᠠ ᠬᠢᠨ ᠠᠰᠠᠷ ᠤᠨ ᠣᠷᠣᠢ ᠶᠢᠨ ᠥᠯ ᠢᠢᠭᠰᠡᠨ ᠥᠨ ᠴᠥᠭ ᠢ ᠬᠡᠮᠵᠢᠭ ᠰᠡᠨ ᠳᠦ 45° ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠲᠤᠰ ᠠᠰᠠᠷ ᠤᠨ ᠦᠨᠳᠤᠷ ᠨᠢ ᠪᠠᠷᠤᠭ ( ) A.8 2m B.1 63 m C.52 m D.7 0m 17 . ᠵᠢᠷᠤᠭ ᠮᠡ ᠲᠤ᠂ ᠲᠠ ᠨ ᠠ ᠥᠯ ᠢᠢᠭᠰᠡᠨ ᠥᠨᠴᠥ ᠭ ᠢ ᠬᠡᠮᠵᠢᠬᠦ ᠬᠡᠮᠵᠢᠬᠦᠷ ᠢᠶᠡᠨ ᠲᠤᠭ ᠤ ᠨ ᠡᠰᠢ AB ᠡᠴᠡ 6m ᠵᠠᠢ ᠲᠠ ᠢC ᠴᠡᠭ ᠲᠦ ᠲᠠᠯᠪᠢ ᠵᠤ᠂ ᠲᠤᠭ ᠤ ᠨ ᠡᠰᠢ ᠶᠢᠨ ᠣ ᠷᠣᠢ 𝐵 ᠶᠢ ᠬᠡᠮᠵᠢᠭᠰ ᠡᠨ ᠳᠦ ᠥᠯ ᠢᠢᠭᠰᠡᠨ ᠥᠨᠴ ᠥᠭ 𝛼 = 60° ᠂ ᠬᠡᠮᠵᠢᠬᠦᠷ ᠦᠨ ᠦᠨᠳᠤᠷ CD ＝ 1m ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ ᠲᠤᠭ ᠤ ᠡᠰᠢ AB ᠶᠢᠨ ᠦᠨᠳᠤᠷ ᠨᠢ ( ᠪᠣᠳᠣᠭᠰ ᠠᠨ ᠳ ᠦ᠋ᠩ ᠳᠦ ᠪ ᠡᠨ ᠢ ᠵᠠᠭᠤᠷ ᠤᠨ ᠲᠡᠮᠳᠡ ᠭ ᠦᠯ ᠡᠳᠡᠭᠡ ) ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠤᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰᠡᠳᠦ ᠪ ᠃ (ᠨᠡ ᠢᠲᠡ 62 ᠬᠤᠪᠢ) 18 .(1 6 ᠬᠤᠪᠢ) ᠳᠠ ᠷᠠᠭᠠᠬ ᠢ ᠨᠥᠬᠥ ᠴᠡᠯ ᠢ ᠦᠨᠳᠤᠰ ᠤᠯᠡᠨ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ ᠢ ᠪᠣᠳᠣ᠃ Rt ∆AB C ᠳᠦ᠂ ∠C = 90° ᠄ (1) ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ᠄ 𝑎= 4᠂ 𝑏= 8 (2) ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ 𝑏 = 10 ᠂ ∠ B= 60° (3) ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ 𝑐 = 20 ᠂ ∠ A= 60 ° (4) ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ 𝑎 = 5᠂ ∠B=3 5° （ 45 30  B A D C 19 . (10 ᠬᠤᠪᠢ) ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠢ ᠯᠡᠷᠭᠡᠢᠯ ᠡᠯ ᠢ ᠪᠣᠳᠣ᠃ (1) sin 230 °+c os 245 °+ √2 sin 60 °·t an 45 ° (2) 𝑠𝑖𝑛 230° :𝑐𝑜𝑠 260° 𝑡𝑎𝑛 60° ×𝑡 𝑎𝑛 30° + 𝑠𝑖𝑛 45° 20 . (12 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠥᠪᠰᠢᠨ ᠭᠠᠵ ᠠᠷ ᠳᠡ ᠭᠡᠷ ᠡ ᠨᠢᠭᠡ ᠰᠠᠭ ᠤᠷᠢ ᠮᠤᠳᠤᠨ ᠤ ᠦᠨᠳᠤᠷ ᠨᠢ 5 ᠮᠧᠲᠷ᠂ ᠠ ᠳᠠᠯᠢ ᠪᠤᠰᠤ ᠬᠤᠭ ᠤᠴᠠᠭ ᠠᠨ ᠳᠤ ᠬᠣᠶ ᠠᠷ ᠤᠳᠠ ᠭ ᠠ ᠮᠤᠳᠤ ᠨ ᠤ ᠰᠡᠭᠦᠳᠡ ᠷ ᠢ ᠠᠵᠢᠭᠯ ᠠᠪᠠ᠃ ᠨᠢᠭᠡᠳᠤᠡᠷ ᠤᠳᠠ ᠭ ᠠ ᠠᠵᠢᠭ ᠯᠠᠬᠤ ᠳ ᠤ ᠨᠠ ᠷᠠᠨ ᠤ ᠭᠡᠷᠡᠯ ᠪᠠ ᠲᠥ ᠪᠰᠢᠨ ᠭᠠᠳᠠ ᠷᠭᠤ ᠶᠢ ᠨ ᠬᠠ ᠪᠴᠢᠭᠰ ᠠᠨ ᠥᠨᠴᠥ ᠭ ᠨᠢ 45 °᠂ ᠬᠣᠶ ᠠᠳᠤᠭ ᠠᠷ ᠤᠳᠠ ᠭ ᠠ ᠠᠵᠭᠢ ᠯᠠᠬᠤ ᠳᠤ ᠬᠠᠪ ᠴᠢᠭᠰ ᠠᠨ ᠥᠨᠴᠥ ᠭ ᠨᠢ 30 °᠂ ᠲᠡ ᠭᠡᠪᠡᠯ ᠤᠳᠠ ᠭᠠᠨ ᠤ ᠠᠵᠢᠭ ᠯᠠᠬᠤ ᠦᠶ ᠡᠰ ᠦ ᠨ ᠰ ᠡᠭᠦᠳᠡ ᠷ ᠨᠢ ᠬᠡ ᠳᠤᠢ ᠮᠧᠲᠷ ᠵ ᠦ᠋ᠷᠢᠬᠦ ᠪᠣᠢ︖ 21 . (1 2 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ AB ᠪᠣᠯ ᠮᠥᠷ ᠡᠨ ᠦ ᠬᠠᠵᠠᠭᠤ ᠳᠠᠬ ᠢ ᠨ ᠢᠭᠡ ᠭᠡᠰᠡᠭ ᠵᠠᠮ ᠂ ᠵᠠᠮ ᠤ ᠨ ᠤᠷ ᠳᠤ ᠨᠢ 3 ᠺᠢᠯᠥᠮᠧᠲᠷ ᠂ C ᠨᠢ ᠮᠥᠷᠡ ᠨ ᠦ ᠨᠥᠬᠥ ᠭᠡ ᠬᠠᠵᠠᠭ ᠤ ᠳᠠᠬ ᠢ ᠨᠢ ᠭᠡ ᠭᠠᠷᠠᠮ᠂ ᠮᠥᠷᠡ ᠨ ᠦ ᠬᠣᠶ ᠠᠷ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠵᠠᠮ ᠬᠠᠷᠢᠯ ᠴᠠᠭᠠᠨ ᠤ ᠠᠰᠠᠭᠤᠳᠠ ᠯ ᠢ ᠰᠢᠢᠳᠪ ᠤᠷᠢᠯ ᠡᠬᠦ ᠶᠢᠨ ᠲᠥ ᠯᠥᠭᠡ C ᠭᠠᠷᠠᠮ ᠠᠴᠠ ᠨᠢ ᠭᠡ ᠬᠥᠭᠥᠷ ᠭᠡ ᠪᠠᠷᠢᠬ ᠤ ᠪᠠᠷ ᠬᠡᠮᠵᠢᠯ ᠲᠡ ᠬᠢᠭᠰᠡᠨ ᠳᠦ A ᠭᠠᠵᠠ ᠷ ᠨᠢ ᠭᠠᠷ ᠠᠮ C ᠠᠴᠠ ᠠᠵᠢᠭ ᠯᠠᠬᠤ ᠳᠤ ᠬᠣ ᠢᠨ ᠠ ᠠᠴ ᠠ ᠪ ᠠᠷᠠᠭ ᠤᠨᠰᠢ 30 ° ᠴᠢᠭᠯᠡᠯ ᠳᠦ ᠪᠣᠢ᠂ B ᠭᠠ ᠵᠠᠷ ᠨᠢ ᠭᠠᠷ ᠠᠮ C ᠶᠢᠨ ᠵᠡᠭᠦᠨ ᠬᠣᠢᠳᠤ ᠴᠢᠭᠯᠡᠯ ᠳᠦ ᠣᠷᠣᠰ ᠢᠨ ᠠ᠂ ᠭᠠᠷ ᠠᠮ C ᠠᠴᠠ ᠮᠥᠷᠡ ᠨ ᠦ ᠬᠣᠶ ᠠᠷ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠵᠠᠮ ᠬᠠᠷᠢᠯ ᠴᠠᠭ ᠠ ᠶᠢ ᠰᠢᠢᠳ ᠪᠦᠷᠢᠯᠡᠬ ᠦ ᠬᠠᠮ ᠤᠭ ᠣᠬ ᠣᠷ ᠵᠠ ᠢ ᠨᠢ ᠬᠡᠳᠤᠢ ᠪᠣᠢ ︖ (0.1 ᠪᠣᠯ ᠲᠠᠯ ᠠ ᠨᠠᠷᠢᠪ ᠴᠢᠯᠠ) 22. (1 2 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠴᠡᠭ A ᠪᠣᠯ ᠷᠠᠳᠢᠤᠰ ᠨᠢ ᠪᠠᠷᠤᠭ 30 0 ᠮᠧ ᠲᠷ ᠪᠣᠯᠬ ᠤ ᠲᠣᠭ ᠣᠷᠢᠭ ᠬᠡᠯᠪᠡ ᠷᠢ ᠲᠠ ᠢ ᠣ ᠢ ᠰᠢᠭ ᠤᠢ ᠶᠢ ᠨ ᠲᠥᠪ ᠪᠣᠯᠣᠨ ᠠ᠃ ᠣ ᠢ ᠰᠢᠭ ᠤᠢ ᠶᠢ ᠨ ᠣ ᠢᠷ ᠠ ᠬᠠᠪᠢ ᠳᠤ B , C ᠬᠣᠶ ᠠᠷ ᠲᠤᠰ ᠬᠤᠨ ᠪᠣᠢ᠂ ᠣᠲᠣ ᠬᠣᠶ ᠠᠷ ᠲᠤᠰᠬ ᠤᠨ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠨᠡᠪᠳ ᠡᠷᠡᠬᠦ ᠶᠢᠨ ᠲᠥ ᠯᠥᠭᠡ᠂ ᠲᠤᠰ ᠬᠤᠨ ᠤ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠤ 1 00 0 ᠮᠧᠲᠷ ᠦᠨ ᠠᠰᠬ ᠠᠨ ᠵᠠᠮ ᠵᠠᠰᠠᠪᠠ ᠯ᠂ ᠬᠡ ᠮᠵᠢᠯ ᠲᠡ ᠪᠡᠷ ∠AB C= 45 ° ᠂ ∠A CB = 30° ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠡᠨᠡ ᠵᠠᠮ ᠨᠢ ᠣᠢ ᠰᠢᠭ ᠤᠢ ᠶᠢ ᠳᠠᠭ ᠠᠷᠠᠬᠤ ᠡᠰᠡᠬ ᠦ ᠶᠢ ᠮᠠᠭᠠᠳᠯ ᠠ᠃ ( ᠮᠠ ᠭᠠᠳᠯ ᠠᠭᠰᠠᠨ ᠶᠠᠪᠤ ᠴᠠ ᠪᠠᠨ ᠲᠣᠳ ᠣᠷᠬᠠᠢ ᠪᠢᠴᠢᠨ ᠡ᠃ ) A B H C