和角公式與差角公式

 


1. 在坐標平面的\(x\)軸上有\(A(\,2\;,\;0\,)\),\(B(\, - 4\;,\;0\,)\)兩觀測站,同時觀察在\(x\)軸上方的一目標\(C\)點,測得\(\angle BAC\)及\(\angle ABC\)之值後,通知在\(D(\,\frac{5}{2}\;,\; - 8\,)\)的砲臺;此兩個角的正切值分別為\(\frac{8}{9}\)及\(\frac{8}{3}\)。那麼砲臺\(D\)至目標\(C\)的距離為__________。
【90推甄】

2. 如圖,\(\Delta ABC\)的對邊分別為\(a\),\(b\),\(c\),\(P\)為\(C\)點的垂足,\(h\)為高,\(BP = x\),\(AP = y\),則下列選項哪些必定為真?
(1) \(\cos C = \frac{h}{a} + \frac{h}{b}\)
(2) \(\cos C = \frac{x}{a} + \frac{y}{b}\)
(3) \(\cos C = \cos (A + B)\)
(4) \(\cos C = \frac{{{a^2} + {b^2} - {c^2}}}{{2ab}}\)
(5) \(\cos C = \frac{{{h^2} - xy}}{{ab}}\) 。
【91學測補】

sa8 16

3. 如圖所示的立體示意圖,線段\(\overline {AC} \)垂直於過\(D\)、\(C\)、\(E\)這三點的平面。設\(\overline {AB} = \overline {BC} = 10\),\(\overline {DC} = 15\),\(\overline {CE} = 30\),\(\angle CDB = \alpha \),\(\angle BDA = \beta \),\(\angle CEB = \alpha '\),\(\angle BEA = \beta '\)。試問下列何者為真?
(1) \(\alpha = \beta \)
(2) \(\alpha = \alpha ' + \beta '\)
(3) \(\alpha = 2\alpha '\)
(4) \(\alpha + \beta > \frac{\pi }{3}\)
(5) \(\alpha ' + \beta ' < \frac{\pi }{6}\) 。
【92乙】

sa8 17

4. \(\Delta ABC\)為邊長為5的正三角形,\(P\)點在三角形內部,若線段長度\(\overline {PB} = 4\)且\(\overline {PC} = 3\),則\(\cos \angle ABP = \)__________(四捨五入到小數點後第二位,\(\sqrt 2 \)的近似值是1.414,\(\sqrt 3 \)的近似值是1.732)。
【98甲】

5. 在\(\Delta ABC\)中,已知\(\overline {AB} = 5\),\(\cos \angle ABC = - \frac{3}{5}\),且其外接圓半徑為\(\frac{{13}}{2}\),則\(\sin \angle BAC = \)__________。(化成最簡分數)
【99甲】

6. 在坐標平面上,廣義角\(\theta \)的頂點為原點\(O\),始邊為\(x\)軸的正向,且滿足\(\tan \theta = \frac{2}{3}\)。若\(\theta \)的終邊上有一點\(P\),其\(y\)坐標為\( - 4\),則下列哪些選項一定正確?
(1) \(P\)的\(x\)坐標是6
(2) \(\overline {OP} = 2\sqrt {13} \)
(3) \(\cos \theta = \frac{3}{{\sqrt {13} }}\)
(4) \(\sin 2\theta > 0\)
(5) \(\cos \frac{\theta }{2} < 0\)。
【101學測】

7. 設\(0 \le \theta < 2\pi \),且方程式\({x^2} - a = 0\)之兩根恰為\(\sin \theta \)與\(\cos \theta \)。請選出正確的選項。
(1) \(\tan \theta = 1\)
(2) \(\sin (\theta + \frac{\pi }{4}) = 0\)
(3) \(\sin 2\theta = - 1\)
(4) \(a = \frac{1}{2}\)
(5) 滿足題設的\(\theta \)只有一個。
【101甲】

8. 設銳角三角形\(ABC\)的外接圓半徑為8。已知外接圓圓心到\(\overline {AB} \)的距離為2,而到\(\overline {BC} \)的距離為7,則\(\overline {AC} = \)__________。(化成最簡根式)
【102學測】

Ans:
1. 13
2. (4)(5)
3. (2)
4. 0.92
5. \(\frac{{33}}{{65}}\)
6. (2)(4)
7. (1)(3)(4)
8. \(4\sqrt {15} \)

 

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sgod 的個人頭像
sgod replied the topic: #616 3 年 8 個月 ago
關於三角函數的問題可以在這快速回覆或者到討論區中的「中學數學」版中討論

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