單元08 基礎題類題
- \({2500^ \circ }\)為第__________象限角,其最小正同界角為__________,最大負同界角為__________。
- __________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。
- 設\(\theta \)為一銳角,已知\(\tan \theta = \frac{1}{2}\),則\(\sin \theta = \)__________,\(\cos \theta = \)__________。
- 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\(\overline {OC} = \)__________。
- 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\(\sin \angle EFC = \)__________。
- (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) \(\sin {240^ \circ } \times \tan {300^ \circ } + \cos {300^ \circ } \times \tan {315^ \circ } = \)__________。
(2) \(\sin {270^ \circ } \times \cos {360^ \circ } + \sin {315^ \circ } \times \tan {360^ \circ } = \)__________。
- (1) \(\sin ({180^ \circ } + \theta ) \cdot \cos ({90^ \circ } + \theta ) - \sin ({90^ \circ } - \theta ) \cdot \cos ({180^ \circ } + \theta ) = \)__________。
(2) \(\tan ({180^ \circ } - \theta ) \cdot \cos (\theta - {180^ \circ }) + \sin (\theta - {180^ \circ }) = \)__________。
- 如下圖,已知\(\angle C = {90^ \circ }\),\(\overline {AB} = 2\),\(\overline {AC} = 1\),則\(\tan \theta = \)__________。
- 設\(\sin \theta = \frac{5}{{13}}\),且\(\tan \theta < 0\),則\(5\sin \theta + \cos \theta = \)__________。
- \({(\sin {20^ \circ } + \sin {70^ \circ })^2} + {(\cos {20^ \circ } - \cos {70^ \circ })^2} = \)__________。
- (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__________。
- 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\( = \)__________。
- 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) \(\tan \frac{{3\pi }}{4} = \)__________。
- 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__________。
- 在\(\Delta ABC\)中,\(\overline {AB} = 3\),\(\overline {BC} = 7\),\(\overline {CA} = 8\),則\(\angle A = \)__________。
- 在\(\Delta ABC\)中,已知\(\overline {AB} = 5\),\(\overline {AC} = 3\),\(\angle A = {120^ \circ }\),則\(\overline {BC} \)的長度為__________。
- 如下圖,\(\Delta ABC\)中,\(\overline {AB} = 10\),\(\overline {BC} = 14\),\(\overline {CA} = 6\)
(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)
- \(\sin {85^ \circ }\cos {50^ \circ } + \cos {85^ \circ }\sin {50^ \circ } = \)__________。
- \(\sin {75^ \circ } = \sin ({45^ \circ } + {30^ \circ }) = \)__________,\(\cos {75^ \circ } = \cos ({45^ \circ } + {30^ \circ }) = \)__________,\(\tan {75^ \circ } = \tan ({45^ \circ } + {30^ \circ }) = \)__________。
- 設\({0^ \circ } < \alpha < {90^ \circ }\),\({90^ \circ } < \beta < {180^ \circ }\),\(\cos \alpha = \frac{3}{{\sqrt {10} }}\),\(\cos \beta = \frac{{ - 1}}{{\sqrt 5 }}\),則
\(\sin (\alpha + \beta ) = \)__________,\(\cos (\alpha + \beta ) = \)__________,\(\alpha + \beta = \)__________。
- 設\({180^ \circ } < \theta < {270^ \circ }\),且\(\cos \theta = - \frac{4}{5}\),則
\(\sin \frac{\theta }{2} = \)__________,\(\cos \frac{\theta }{2} = \)__________,\(\tan \frac{\theta }{2} = \)__________,
\(\sin 2\theta = \)__________,\(\cos 2\theta = \)__________,\(\tan 2\theta = \)__________,
\(\sin 3\theta = \)__________,\(\cos 3\theta = \)__________。
- 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__________。
- 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 |
- 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)
- 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. 1
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. 7
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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