單元08 基礎題類題

  1. \({2500^ \circ }\)為第__________象限角,其最小正同界角為__________,最大負同界角為__________。
  1. __________若有向角\(\theta \)為第二象限角,則\(\frac{\theta }{2}\)可能為第幾象限角?(多選) (1) 一  (2) 二  (3) 三  (4) 四。
  1. 設\(\theta \)為一銳角,已知\(\tan \theta = \frac{1}{2}\),則\(\sin \theta  = \)__________,\(\cos \theta  = \)__________。
  1. 設圓\(O\)之半徑為5,\(\overline {OC} \)交圓\(O\)於\(A\)點,\(\overline {CD} \)切圓\(O\)於\(D\)點,\(B\)為\(A\)點到\(\overline {OD} \)的垂足,\(\overline {AB} = 3\),如下面的示意圖。則\(\overline {OC}  = \)__________。
    08 04
  1. 如下圖所示(只是示意圖),將梯子\(\overline {AB} \)靠在與地面垂直的牆\(\overline {AC} \)上,並測得梯子長\(\overline {AB} = 120\)公分,且\(\overline {AB} \)與水平地面的夾角\(\angle ABC\)為\({60^ \circ }\)。將在地面上的底\(B\)沿著地面向外拉30公分到點\(F\)(即\(\overline {FB}  = 30\)公分),則梯子\(\overline {EF} \)與地面的夾角\(\angle EFC\)之正弦值為\(\sin \angle EFC = \)__________。
    08 05
  1. (1) \(\sin {30^ \circ } \cdot \cos ( - {150^ \circ }) - \sin ( - {120^ \circ }) \cdot \cos {60^ \circ } = \)__________。
    (2) \(\sin {0^ \circ } \times \sin {15^ \circ } + \cos {90^ \circ } \times \cos {135^ \circ } + \tan {45^ \circ } \times \tan {135^ \circ } = \)__________。
  1. (1) \(\sin {240^ \circ } \times \tan {300^ \circ } + \cos {300^ \circ } \times \tan {315^ \circ } = \)__________。
    (2) \(\sin {270^ \circ } \times \cos {360^ \circ } + \sin {315^ \circ } \times \tan {360^ \circ } = \)__________。
  1. (1) \(\sin ({180^ \circ } + \theta ) \cdot \cos ({90^ \circ } + \theta ) - \sin ({90^ \circ } - \theta ) \cdot \cos ({180^ \circ } + \theta ) = \)__________。
    (2) \(\tan ({180^ \circ } - \theta ) \cdot \cos (\theta - {180^ \circ }) + \sin (\theta  - {180^ \circ }) = \)__________。
  1. 如下圖,已知\(\angle C = {90^ \circ }\),\(\overline {AB} = 2\),\(\overline {AC}  = 1\),則\(\tan \theta  = \)__________。
    08 09
  1. 設\(\sin \theta = \frac{5}{{13}}\),且\(\tan \theta  < 0\),則\(5\sin \theta  + \cos \theta  = \)__________。
  1. \({(\sin {20^ \circ } + \sin {70^ \circ })^2} + {(\cos {20^ \circ } - \cos {70^ \circ })^2} = \)__________。
  1. (1) 直角坐標系中,點\((\, - 2\sqrt 3 \;,\; - 2\,)\)的極坐標表示法為__________。
    (2) 極坐標系中,點\([\,10\;,\;{300^ \circ }\,]\)的直角坐標表示法為__________。
  1. 將下列角度換成弧度,或將弧度換成角度:
    (1) \({150^ \circ } = \)__________。
    (2) \( - {300^ \circ } = \)__________。
    (3) \(\frac{{7\pi }}{6}\)弧度\( = \)__________。
    (4) \( - \frac{{7\pi }}{4}\)弧度\( = \)__________。
  1. 請計算下列三角函數的值:
    (1) \(\sin \frac{{4\pi }}{3} = \)__________。
    (2) \(\cos \left( { - \frac{{5\pi }}{6}} \right)\)=__________。
    (3) \(\tan \frac{{3\pi }}{4} = \)__________。
  1. 在\(\Delta ABC\)中,\(\angle B = {45^ \circ }\),\(\angle C = {30^ \circ }\),\(\overline {AC} = 6\sqrt 2 \),則\(\overline {AB} \)之長為__________,外接圓半徑為__________。
  1. 在\(\Delta ABC\)中,\(\overline {AB} = 3\),\(\overline {BC}  = 7\),\(\overline {CA}  = 8\),則\(\angle A = \)__________。
  1. 在\(\Delta ABC\)中,已知\(\overline {AB} = 5\),\(\overline {AC}  = 3\),\(\angle A = {120^ \circ }\),則\(\overline {BC} \)的長度為__________。
  1. 如下圖,\(\Delta ABC\)中,\(\overline {AB} = 10\),\(\overline {BC}  = 14\),\(\overline {CA}  = 6\)
    08 18

    (1) 設\(\overline {AH} \)為\(\overline {BC} \)邊上的高,則\(\overline {AH} \)之長為__________。
    (2) 設點\(M\)為之中點,則中線\(\overline {AM} \)之長為__________。
    (3) 設\(\overline {AD} \)為\(\angle A\)之角平分線,則\(\overline {AD} \)之長為__________。(請參考:穩拿複
       習講義p.219 範例8)
  1. \(\sin {85^ \circ }\cos {50^ \circ } + \cos {85^ \circ }\sin {50^ \circ } = \)__________。
  1. \(\sin {75^ \circ } = \sin ({45^ \circ } + {30^ \circ }) = \)__________,\(\cos {75^ \circ } = \cos ({45^ \circ } + {30^ \circ }) = \)__________,\(\tan {75^ \circ } = \tan ({45^ \circ } + {30^ \circ }) = \)__________。
  1. 設\({0^ \circ } < \alpha < {90^ \circ }\),\({90^ \circ } < \beta  < {180^ \circ }\),\(\cos \alpha  = \frac{3}{{\sqrt {10} }}\),\(\cos \beta  = \frac{{ - 1}}{{\sqrt 5 }}\),則
    \(\sin (\alpha  + \beta ) = \)__________,\(\cos (\alpha  + \beta ) = \)__________,\(\alpha  + \beta  = \)__________。
  1. 設\({180^ \circ } < \theta < {270^ \circ }\),且\(\cos \theta  =  - \frac{4}{5}\),則
    \(\sin \frac{\theta }{2} = \)__________,\(\cos \frac{\theta }{2} = \)__________,\(\tan \frac{\theta }{2} = \)__________,
    \(\sin 2\theta  = \)__________,\(\cos 2\theta  = \)__________,\(\tan 2\theta  = \)__________,
    \(\sin 3\theta  = \)__________,\(\cos 3\theta  = \)__________。
  1. 下圖是由三個直角三角形堆疊而成的圖形,若\(\overline {AB} = 8\),則直角三角形\(OAB\)的高\(AB\)為__________。
    08 23
  1. 由底下三角函數值表可查得\(\cos {27^ \circ }20' \approx \)__________,\(\cos {62^ \circ }10' \approx \)__________。

角  度

\(\sin \)

\(\cos \)

\(\tan \)

   

27°

00΄

.4540

.8910

.5059

63°

00΄

 

10΄

.4566

.8897

.5132

 

50΄

 

20΄

.4592

.8884

.5169

 

40΄

 

30΄

.4617

.8870

.5206

 

30΄

 

40΄

.4643

.8857

.5243

 

20΄

 

50΄

.4669

.8843

.5280

 

10΄

28°

00΄

.4695

.8829

.5317

62°

00΄

 

10΄

.4720

.8816

.5354

 

50΄

 

20΄

.4746

.8802

.5392

 

40΄

 

30΄

.4772

.8788

.5430

 

30΄

 

40΄

.4797

.8774

.5467

 

20΄

 

50΄

.4823

.8760

.5505

 

10΄

29°

00΄

.4848

.8746

.5543

61°

00΄

   

\(\cos \)

\(\sin \)

\(\cot \)

角  度

  1. 已知\(\sin {36^ \circ }50' \approx 0.5995\),\(\sin {37^ \circ } \approx 0.6018\),若\(\sin \theta = 0.6\)且\({0^ \circ } < \theta  < {90^ \circ }\),則由內插法可得\(\theta  \approx \)__________。(四捨五入至單位分的個位數)
  1. 某人隔河測一山高,在\(A\)點測量山時,山的方位為東偏北\({60^ \circ }\),山頂的仰角為\({45^ \circ }\),某人自\(A\)點向東行600公尺到達\(B\)點,山的方位變成在西偏北\({60^ \circ }\),則山有多高?__________公尺。【91學測】

Ans:

1. 四,\({340^ \circ }\),\( - {20^ \circ }\)

2. (1)(3)

3. \(\frac{{\sqrt 5 }}{5}\),\(\frac{{2\sqrt 5 }}{5}\)  

4. \(\frac{{25}}{4}\) 

5. \(\frac{{\sqrt 7 }}{4}\)  

6. (1) \(0\) (2) \( - 1\) 

7. (1) 1  (2) \( - 1\) 

8. (1) 1  (2) 0  

9. \( - \sqrt 3 \) 

10.

11. 2

12. (1) \([\,4\;,\;{210^ \circ }\,]\) (2) \((\,5\;,\; - 5\sqrt 3 \,)\)

13. (1) \(\frac{{5\pi }}{6}\) (2) \( - \frac{{5\pi }}{3}\) (3) \({210^ \circ }\) (4) \( - {315^ \circ }\)

14. (1) \( - \frac{{\sqrt 3 }}{2}\) (2) \( - \frac{{\sqrt 3 }}{2}\) (3) \( - 1\)

15. 6,6

16. \({60^ \circ }\)  

17.

18. (1) \(\frac{{15\sqrt 3 }}{7}\)

(2) \(\sqrt {19} \) (3) \(\frac{{15}}{4}\) 

19. \(\frac{{\sqrt 2 }}{2}\) 

20. \(\frac{{\sqrt 6  + \sqrt 2 }}{4}\),\(\frac{{\sqrt 6  - \sqrt 2 }}{4}\),\(2 + \sqrt 3 \)

21. \(\frac{{\sqrt 2 }}{2}\),\( - \frac{{\sqrt 2 }}{2}\),\({135^ \circ }\)

22. \(\frac{3}{{\sqrt {10} }}\),\( - \frac{1}{{\sqrt {10} }}\),\(\frac{{24}}{{25}}\),\(\frac{7}{{25}}\),\(\frac{{24}}{7}\),\( - \frac{{117}}{{125}}\),\(\frac{{44}}{{125}}\)

23. 32

24. 0.8884,0.4669

25. \({36^ \circ }52'\) 

26. 600

 

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