1969-01-07 — Page 15

育教伊華真三第張四第二日九十月一十年申戊醫夏 WAH KIU YAT PC

報日僑華

二期星

文中學會考試題預習

數學科

(十)

「歐陽鐺文·

ATHEMATICS (10)

LESSON 10: RATIOS OF ANGLES OF ANY MAGNITUDE

SINE & COSINE FORMULAE FOR

The trigonometrical ratios of any angle, at pt. P are defined as

·Sint-

y-ordinate of 1

OP.

Cose A-erdinate of

-ordinate of P PN

x-ordinate ar P

with centre O, fixed radius Oy, let a circle be described then the xsy axes divide the circle into 4 quadrants. If F,P,P,F4 be the positions of P in the four quadrants as shown, then it will be seen that (rading OP 14 always. "be regarded as positive fer OF only fixes. the boundary

sf 0);

(1)In the 1st quadrant:All the values are 4)ve. (1)In the 2nd quadrant : Only the x erdinate is -lve. 111)In the 3rd quadrant:Both the x & y ordinates are (1)In the 4th quadrant Only the y-ordinate is -ve. fence, we find:

sine & casecant

cosine & secant

tangent & CE

(2) ■* = b2 + c2- 2bc ces A (cesine formula) (3) bcose + c cos B

han any three parts (at least one of them must be à 9100; of a triangle are given, the triangle can be seived by ene of the above showed relations. Use cosine law for the exp, to the 2nd degree (1.6. squared). The cases,

such as AAS, ASA, and SSA can be served by sine law..

Given: Note: An ambiguous case only exist in SSA. e.g.

a, b; if b> a>b-sin A and A≤ 90", then it is an #mbiguone case.

(a)In & ABC:(1)ginAsimo

-then knews 2B.

(11) 40 =180° - ( LA+ZB)=known.

(iii) and

that kriews

(b) In 4 AB1C (1) ZB1=180–48 = known

(11) 4C = 180°- (4A=48)= known.

(111) SMC Sin AC is known

SAMPLE 4: Two sides of a triangle are:

Hand.

solve the trim

and the included angle is 60 angles and find the area of the triangle, and Pind the area of the triangle. SOLUTION: Denets the 4 by ABC

b

ig: cosine fermiila,

·A=60"

(8+415)

I've

+'ve

've

All the fundamental formulas, such as

sin2m+cosx=1; 1+tan Kased d tan eld in all cases. (1.e. true for any oc)

t've

O've

Formula.

Sin 8-ban A

Sin-a

Sin C

Sin A

caso

EXAMPLE 1: If tan A --, find sin SOLUTION: The boundary line of A, an

shown, will lie either in tha 2nd or 4th quadrant.

(1.0 GP OPE)

either case. the raddins Offic

From 2nd ques

ein A Ford +15

from kth quadrant; sin

OP

and cos

Jon=16; COSA

NE

Note: There are two darrespending values for both ain.

and cos A. If, however, it is known in which quad- rant the boundary line of Aldes, sin À & cheA have each a single value.

From the ahove definitions, we find that PN &Oy changw. aa P moves round the circle, Hence, the variations of the trigonometrical' functions if the angle, e, depend on the position of P. As a result, we find:

-By Lieum

105

0.2588

(2A+∠B)

(1) Starty whenever possible, by finding the

smallest angle which cannot be obtuse,

(2) Calculate the angle opposite to the smaller

of the given sides, which cannet be ebtuse. (3) A a check, greater side is always opposite

te greater angle. (and also the converse). And, a triangle cannet has 2 ebtusa za

KXAMPLE...» STwo angles er a triangular field are 224 and

45 and the length of the side opposite te the later is one furlong find the area.

SOLUTION; Denote: the triangular field by △ABC,

sit. 4A 22, 48#45°

bal furlang=220ya..

By LaSua af A: 40= 180′′- (GAL4B)

· 180° - ( 224* ~45′): 1121

Where:

SINE

COSINE

increasing an absolute value decreasing.

EXAMPLE 2: Find the value of

TANGENT

From sine formula,

sin 224 ein 224 sin

sin A ein B

sin 45

ya. and

ain C

sin 112)

:220 sín 1124*

sin 450.

6日七月二十年九六九一圈公年七十五國民華中

「以中文大學爲主。

才相進入大器攻度的說法。.

·等教育,打破過去傳調估,只有蓋子第 數字顯示許音遹家無的子弟,都有进痧造人高

·又識新生平均成崴能月收入緒計數字,在一 一千元以下收入的家糕佔總數百分二六·五。上述

(E)

院書亞新學大文中 生新冊註年學本

生學中文中爲多

華

教

育

中學生就業工界問題

·李思義、

供應稅未-

份,此其難 工作塲所,編成之工業師賽,此三者現在本港之日 (星期日)與實 人类执之工業中華,必須有資質之設倍,寫能之一,故感,將於元月1 .絕不能追上本港社會之然求,况旦開或一所能視各委員爲結會

實代表学患雖在會上以「中華生就無工界隨孃」 合之處,並大大影响中學畢業生就業區題;本 「用顯,独表演舞全文如下: 開始時已鄭明文法申華生,不組立刻拉萊工界面,十,都有 一男教育的出路校兔界一在工碘界方面,就王希望教育界給與波等一套遊新界壯·迎 刘可用之技工,不須再行訓練,,此見隊亦有不同葬機在參加。遊方券

校,到時的

,仍與戰前無大差別,而牠會人士對於中繞著了,是美校,並非工廠,而工獲學校基生,並不等淤船進程序如下,上午

· 第二六世界大破後,若泡轉變爲工歲壯會,「不須練,即使工業學校攝樂生,亦不能於艦校不佔位者海位十元, 名額限一百二十張

對中文大學學生會會章事

一位同學不同意校方意見

甘之

備全之現象,必須消除

出八十九大,佔總收百分][ 小點+英文中學會考中 百四十人,佔總數百分六

挑計中,核兮新生大多數

·環被校對新生之調查

,别在該校千三等食讖一年

找的先

漢一入,佔總數百分點酷

升本抛不眼工

產生的?他們代表了多少同學了,

以中色 智行器四年多了,還不能成立「中大學生會」? .. 所學生代表是識,间孝們根本不知道何事,那些代表深是

·佈共組織及組員人選?佰威什麼「養上甜利姿員會」,那些

編輯先生:上月底好在竞秀君到中大數十位同學 一,校方之意,認爲同學們所聽「草案」由柳方定具 ,,B此「委員只」組織了這些日子,以「想從女协同畠公一下許多,所以不能及引中學車業生。其有,本港 賞比白掃工作路爲實也,他 之嬴練,以求鼻部 山有三間學院學生代表成員。對此,本人有啟點疑問灣學生參加日領工作,則只須短期之訓練,若參加,以營業之信心,並比學徒較高之得議。 段校方的發言。但本人對校方所言,太無阻意,故寫此倍, 飛則就照,則雖有基本學識,但不能不在湖三,並搴欧誤區,使一戰學生更能禛康工業時代。 A此「委員會」自一九六四年成立至今,做了些什麽?「工業,則狆從頭目學徒始跑不可;而丁感可能枪 五〕家必須改其念,就椅子女就 澳解,實際上是由「生福利委員會」所疾,而此「漫員會」工,木工,證械等科目,眞絕無僅有,就艾糍中一阵,工媵界必須顧工業中學畢業生,與 關謀,凡有意參加者, ̇类論「中大學生會育案」一事,百杨本月一日至覲到一 生,其希望樁進入大學,以接受專業劇線,如進·CF3.現行文法中巫應並教培說工業科目,一俱樂部,四海四十分谈 首先,本著中華絕大部份是文法中學,其 立之目的,是粉與學生以登才激育,並不軟發最 生專門技能,以利其就業,故就讀文法中學之學,行工廠型投入才交流着法。 且本卷交出中學,甚少工科育之,師金機程方面,必須顧及一般工廠之需求

,仍拘殿前之見解,忽塔木活戰後之群與來工廠學徒,萬生在學校接受一般詁本訓練,盲 于生死之報稱,自不怨太衡,比鱀白領工作汝低,工蔡似粉,及就棄工界,城以現在拔工精雕 聖保羅書院同學會

「台,政府爆加緊康股工樂中展,在點借及一小時返回尖沙咀戲除。

·樊會以上意月,本人嘆作下列連丽 ,二時山酒店, 六一,政府赝大趾訓練工檗學校師資,伏一時半臂壓仙蔥,三時代 | 隈向下列地點博尔。

| 当日來各同學購花路。

水湖,六時沖田開源,

「落馬洲,西時期人鄉村

從不必爆

學

*舍能須告徒,

奠定行事

艺校工

之

水

在

餃

非

版

亻

麻

正程

說

(一)雲咸街七十三號 西 建同學會會所,電話

劉兆蒸同學,總仕道中

於一四四宮,體

分則大想望,若家境許可,必媽讓交出出擊。 掌法中馨;若升中試後,子女煞入工業 學者

[十二日新界大旅行

HK 五月 (4)

一期,模方出此言,本人實不效苟同,爲欲呜其不不,請撥出獨貴個一出,

歡迎同携眷參加

TRTECEDORK

「登出,傳近會人士評論之。先致,抖安 临大一學生盛上 一月五日

政府,或不牢利之社團,故此工办學校開敘冶多。路同學會自濬入興街

Sin 46

the area fermula, area at the triangular field 181 主ac. Sim B

320 sin 23Ė) (320 sin 112 ) Sin 46° 7/

squares

sin 245.

((sino + cas @)^4 ( Cas 8 - Sino)"

110 X 220 sin 22±•Sin 67±**

Sin: 45

32100 sqiyd.

(a) 3 sin 0 sec 180 - 2 csc 90. b) 2 sect * cosa - 3 sin3 31

cos.: 360

SOLUTION: (a) 3. sin 0 sec 180+ 2 csc 90′′- cas

• ====3(0)(−1)+2(1)-1

02]

(b) 2 són cos 0 3 sin

-2(-1)(1) + 3(−1) = (1)

3+

The following is a summary of the resulta established:

(a) sin(180°40",

sin(360+

ain( 90eb

(b) cos(180

*cos(36*

+ sin or

tain e or.

cos e er

cos(90 ± 8 )« I.

:) tan(180Y

Lan(360° ± e

tan

tan( 90° ± 6* )x + cot 0

sin(**)* sinc

#sinot

tan( met ok) = # tanec tan(ax) = 1 tanec

EXAMPLE 3: As ahown in fig., PG=Qk, prave. that,

cot ##1(cetar cet y

From: From rt, A OPN:

Adding

cot

From t. ORN: cot

PN RN.

ON (PM+RN.)

FON (2-QN);

2 cot P

Pa=08

Meet Ba(ceta + cot).

In any 4 ABC, (1)

(aine formula

Sin A Sin B. Sin

If the additional relation: sin 45 =2 ein 224 cos 221, is given, then the expression becomes 110 X 120 5.4. NOTE: By subst. the aine formula, sma

Sin B. Sin CA

- ac. sin B

< A sinc].

C

Sin 8.

HINTS & ANS. To Ex. 9

Exp, = 3(#)2 + $(2)-2018-3/4)+4(2)

180 X 66

10800

37800

10800

the area formula becomes;

b sik Ay(_bein (-) sin 8

sina

sine

_b* sin A sinc) or ± [b2 sin A sin C.

SIN (ATC))

In the above example, we can directly find the area by

trom mample 5 Kesson 9), we Radin meaSUM

・subtending

radius

The saga multiplier

EXERCISE 10

Find the vaLUS SI

(a) 4 ces 0 sin 270-3 cos 180 tan 45 (b) 5 tan ces- 800 21

學

2). As shen in fig. 2, AD is a median of OANG, us

the fact preved in Example 3, to express cot

in terms of B, C, and evalute

O in the following casess

(a) 48=35, 40=50°;

(b) LB35° (c)∠B=50

40=1057:

¥35.

3) In AABC, if厶Baire

solve the triangle.

4) Ina sides weiangiovre ats,K+30+3. ahaw that the greatest angle is 120*.

wwwwwwwpha

vertical

mast fram two points A, B on the same level as its Lost H are 32° 14 and 22 28! resp. If the height / of the mast is 52 ft. and the bearings of A and B from H are 270 and 220 resp. find the distance AB and the bearing of B from A.

sum of

each ext angle

raodians

zach im angle

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