ΤΡΙΓΩΝΟΜΕΤΡΙΚΟΙ ΑΡΙΘΜΟΙ
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A √Ì¿‰· 1. N· ·Ô‰Â›ÍÂÙ fiÙÈ: ·) ËÌ137Æ = ËÌ43Æ, ‚) Û˘Ó95Æ = –Û˘Ó85Æ, Á) ÂÊ100Æ = –ÂÊ80Æ. 2. N· ˘ÔÏÔÁ›ÛÂÙ ÙÔ˘˜ ÙÚÈÁˆÓÔÌÂÙÚÈÎÔ‡˜ ·ÚÈıÌÔ‡˜ ÙˆÓ ÁˆÓÈÒÓ: ·) 150Æ, ‚) 103Æ, Á) 175Æ. 3. N· ˘ÔÏÔÁÈÛÙ› Ë ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: ∞ = 2 ËÌ23Æ + Û˘Ó32Æ + Û˘Ó148Æ – ËÌ157Æ, μ = ÂÊ78Æ + Û˘Ó121Æ + Û˘Ó59Æ + ÂÊ102Æ, ° = ËÌ55Æ + ÂÊ135Æ ËÌ125Æ. 4. N· ·Ô‰Â›ÍÂÙ fiÙÈ: ·) Û˘Ó 2 135Æ+ ËÌ150Æ = 1, ‚) Û˘Ó 2 120Æ– ÂÊ 2 150Æ= – . 5. ∞Ó 0Æ ≤ x < 90Æ, Ó· Ï˘ıÔ‡Ó ÔÈ ÂÍÈÛÒÛÂȘ: ·) 2ËÌx – 3 = 0, ‚) 2Û˘Óx – 1 = 0, Á) ÂÊx – 1 = 0, ¯ˆÚ›˜ ÙË ‚Ô‹ıÂÈ· ÙÚÈÁˆÓÔÌÂÙÚÈÎÒÓ ÈӿΈÓ. 6. ∞Ó 0Æ ≤ x < 90Æ, Ó· Ï˘ıÔ‡Ó ÔÈ ÂÍÈÛÒÛÂȘ: ·) –5ËÌx + 4 = 0, ‚) 8Û˘Óx – 3 = 0, Á) 5ÂÊx – 8 = 0. 7. ∞Ó 0Æ ≤ x ≤ 180Æ, Ó· Ï˘ıÔ‡Ó ÔÈ ÂÍÈÛÒÛÂȘ: ·) 5Û˘Óx + 2 = 0, ‚) 3ËÌx – 1 = 0, Á) 6ÂÊx – 21 = 0. 8. ∞Ó 90Æ < x ≤ 180Æ, Ó· Ï˘ıÔ‡Ó ÔÈ ÂÍÈÛÒÛÂȘ: ·) 10Û˘Óx + 3 = 0, ‚) 8ËÌx – 7 = 0, Á) 10ÂÊx + 6 = 0. 9. ∞Ó 90Æ < x ≤ 180Æ, Ó· Ï˘ıÔ‡Ó ÔÈ ÂÍÈÛÒÛÂȘ: ·) 4Û˘Ó 2 x + 3 = 0, ‚) 2ËÌ 2 x – 1 = 0, Á) ÂÊ 2 x – 3 = 0. 10. °È· ÙȘ ÙÈ̤˜ Ù˘ ÁˆÓ›·˜ ˆ Ô˘ ÔÚ›˙ÔÓÙ·È ÔÈ ·Ú·ÛÙ¿ÛÂȘ, Ó· οÓÂÙ ÙȘ Ú¿ÍÂȘ: ∞ = , μ = . 11. ¡· ·Ô‰Â›ÍÂÙ fiÙÈ: ËÌ(90Æ– ˆ) ËÌ(180Æ– ˆ) + Û˘Ó(90Æ– ˆ) Û˘Ó(180Æ– ˆ) = 0. ÂÊ(180Æ– ˆ) Û˘Ó(90Æ– ˆ) Ë̈ Û˘Ó(180Æ– ˆ) ËÌ(90Æ– ˆ) Û˘Óˆ Âʈ ËÌ(180Æ– ˆ) ËÌ(90Æ– ˆ) Ë̈ Âʈ Û˘Ó(180Æ– ˆ) Û˘Ó(90Æ– ˆ) Û˘Óˆ ÂÊ(180Æ– ˆ) 1 12 ∞™∫∏™∂π™ TÚÈÁˆÓÔÌÂÙÚ›· 334 MÂϤÙËÛ˜ ·ÚÎÒ˜ ÙȘ Ï˘Ì¤Ó˜; ∞§ÀΔ∂™
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ΤΡΙΓΩΝΟΜΕΤΡΙΑ ΓΥΜΝΑΣΙΟΥ
Transcript of ΤΡΙΓΩΝΟΜΕΤΡΙΚΟΙ ΑΡΙΘΜΟΙ
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1. N :) 137 = 43, ) 95 = 85, ) 100 = 80.
2. N :) 150, ) 103, ) 175.
3. N : = 2 23 + 32 + 148 157, = 78 + 121 + 59 + 102, = 55 + 135 125.
4. N : ) 2135+ 150 = 1, ) 2120 2150= .
5. 0 x < 90, :) 2x 3 = 0, ) 2x 1 = 0, ) x 1 = 0, .
6. 0 x < 90, :) 5x + 4 = 0, ) 8x 3 = 0, ) 5x 8 = 0.
7. 0 x 180, :) 5x + 2 = 0, ) 3x 1 = 0, ) 6x 21 = 0.
8. 90 < x 180, :) 10x + 3 = 0, ) 8x 7 = 0, ) 10x + 6 = 0.
9. 90 < x 180, :) 42x + 3 = 0, ) 22x 1 = 0, ) 2x 3 = 0.
10. ,
: = ,
= .
11. : (90 ) (180 ) + (90 ) (180 ) = 0.
(180 ) (90 ) (180 ) (90 )
(180 ) (90 ) (180 ) (90 ) (180 )
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12. . .
13. 8 cm , = 3 cm. , .
14. 0 x 90, (3x 2)2 = 4.
15. :) (27 ) = (153+ ), ) (45 ) (135+ ) = 0.
16. 0 x 90, x (x + 30) = (2x + 15).
17. 0 x 180, x = x.
18. 0 x 180, 42x + 2x = 0.
19. 0 x 180, 32x + 2x 1 = 0.
20. 0 x 180, 22x 3x 2 = 0.
21. : ) (90+ ) = , ) (90+ ) = .
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: 1. , 2. , 3. , 4. , 5. .
1. .
2. ) 150 = , 150 = , 150 = ,
) 103 = 77 0,974, 103 = 77 0,225, 103 = 77 4,331,) 175 = 5 0,087, 175 = 5 0,996, 175 = 5 0,087.
3. = 23 0,391, = 0, = 0.4. .5. ) x = 60, ) x = 60, ) x = 45.6. ) x 53, ) x 68, ) x 58.7. ) x 114, ) x 19 x 161, ) x 74.8. ) x 107, ) x 119, ) x 149.9. ) , ) x = 135, ) x = 120.
10. = , = .
11. .12. .
=
=
= 180
= 180
.13. , = 1 cm, = 4 3 cm, = 7 cm,
=
= ,
= ,
= ,
= 43
= 43.
14. x = () x = 0, x = 90.
15. .16. (x + 30) = (2x + 15) (90 x 30) = (2x + 15) 90 x 30 = 2x + 15,
x = 15.17. x = 135.
18. 42x + 2x = 0 2x (2x + 1) = 0, x = 0 x = , x = 90 x = 120.
19. x, x = , x = 1 x = , x = 180
x 71.
20. x, x = , x = 2 () x = ().
21. ) (90 + ) = (90 ) = , ) (90 + ) = (90 ) = .
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