# Quod Erat Demonstrandum

## 2020/02/28

### 正弦積

$\tan 1^o\tan 2^o\tan 3^o\dots \tan 88^o\tan 89^o$

$\tan \theta \tan (90^o-\theta) \equiv 1$

$\tan 1^o\tan 2^o\tan 3^o\dots \tan 88^o\tan 89^o$
$=(\tan 1^o\tan 89^o)(\tan 2^o\tan 88^o)\dots (\tan 44^o\tan 46^o)\tan 45^o$
$=1\times 1\times \dots \times 1$
$=1$

$\sin 1^o\sin 2^o\sin 3^o\dots \sin 88^o\sin 89^o$

(more…)

## 2018/11/20

### 費氏講

Filed under: Additional / Applied Mathematics,Fun,Junior Form Mathematics — johnmayhk @ 6:36 下午
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N 年前往中一班代堂，必談「64 = 65」謎題：

(圖片來源：https://i.stack.imgur.com/fWdMd.jpg)

## 2018/01/23

### 某類三角恆等式記法

Filed under: mathematics,NSS — johnmayhk @ 10:24 下午
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$\sin(-\theta)\equiv -\sin\theta$
$\cos(-\theta)\equiv \cos\theta$
$\tan(-\theta)\equiv -\tan\theta$

## 2017/11/08

### 作正五邊形

Filed under: Additional / Applied Mathematics,Fun,Junior Form Mathematics — johnmayhk @ 10:49 上午
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## 2017/10/28

### 一題多解

Filed under: Junior Form Mathematics,mathematics,NSS — johnmayhk @ 12:14 上午
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（不過有多少學生解完題目，會如此神心尋求另外解法？面對極度規範化的考題，方法多數固定，對一些同學來說，莫說一題多解，更多時是找不到解法。）

## 2017/06/10

### 正多邊形方程

Filed under: Additional / Applied Mathematics,Fun,mathematics,NSS — johnmayhk @ 12:24 下午
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https://www.desmos.com/calculator/vv7stc4nl0

## 2016/12/11

### 小心出題

Filed under: mathematics,NSS — johnmayhk @ 11:18 下午
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$AB=\sqrt{30^2+45^2-2(30)(45)\cos(60^o-40^o)}=19.7$ m

$AB=45\cos 40^o-30\cos 60^o=19.5$ m

## 2016/06/14

### 正七邊形

Filed under: Fun — johnmayhk @ 12:07 上午
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（習題）從片中某些結果不難推出以下命題： (more…)

## 2016/03/21

### 用Ｄ證trigo

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 11:13 上午
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Prove the following identity

$\cos^2x+\cos^2(x+y)-2\cos y\cos x\cos(x+y)=\sin^2y$.

## 2015/10/11

### 正四面體，兩個夾角

Filed under: Fun — johnmayhk @ 9:10 下午
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1.一個玩具

## 2015/10/07

### 三角題

Filed under: NSS — johnmayhk @ 2:58 下午
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Just come across a part of a core mathematics trigonometry problem today morning, I and my student used different ways to solve it, but actually, same results turn up:

Given

$AC=b$,

$CB=a$, where $a > b$

$\angle CAB=\theta < 90^o$,

find $c$ in terms of $a,b,\theta$.

(Note: this is an ‘A.S.S.’ given, but the triangle can be uniquely determined.)

Method 1 (more…)

## 2015/08/03

### 小學三角題

Filed under: Junior Form Mathematics — johnmayhk @ 4:34 下午
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Whatsapp 傳來同事阿仔的暑期功課：在釘板上圍出一個等邊三角形

## 2015/05/20

### 幾何老題

Filed under: Fun — johnmayhk @ 1:40 下午
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$x$ 的大小。（圖有誤，$\angle ACD=30^o$，非 $60^o$

## 2015/05/19

### 某中三三角學題

Filed under: Junior Form Mathematics — johnmayhk @ 11:47 上午
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Given that $\tan \theta=\sqrt{2}$, where $\theta$ is an acute angle. Using trigonometric identities, find the value of $\sin^2\theta - \cos^2\theta$.

## 2014/12/19

### 某些M2題

Filed under: NSS — johnmayhk @ 9:30 下午
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Q.1

$\sin\theta+\cos\theta=\frac{7}{3}$　時，同學已指出：冇可能！因為 $\sin\theta$$\cos\theta$ 的最大值不過是 1。

Q.2