頁二第張六第 日六十月十年西巴圈夏
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MATHEMATICS (4):
四)
Radian Messure: The angle subtended at the centre of
a circle by an are of the circle equal in length to its radius is known as, a radian.
Now the angle subtended by the circumference at the centre of a circle is one complete turn, 1.8. 360. Since the circumference of a circle is 21 times its radius it will subtend 2 radians (2x°) at its centre.
27 radians - 360°
報日僑華
二期星日五廿月一十年九六九一曆公年八十五國民華中 育華
* PQ 2PN - 2 x 5 ein 68°45
10 x 0.9320
9.32 om
(279). Find the
Example 4. The development of a cone is a sector of a cirole of radius r and angle semi-vertical angle of the cone.
Let x be the radius of the cone
e semi-vertical angle of the cone,
Blant edge of the gone
10英文中學會考試題預習專欄
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iwers to Exercise
1. d) out
2. d) down
4. b) off.
e) down upon
(四
ENGLISH LANGUAGE (4)
miniatures quilted
16. o) a generous man.
17. c) checks enthusiasm
18, b) a foolish wild adventure
19. d) the full details
20. a the whole
21. d) had not been
22. c) ought to have gone
23. a) Sitting on the chair
24. b) Having serached the rooms 25. a) them was
5. a) on
d) with
7. a) concealment.
dis pel
9.
fra cas
10.
Circumference of the base of the cone are of the
∙11.
26. a) is
Rector
12, c) landmark
27. a was
13. a) gastrië
28. a ja
14. b) needy
29. d) was
30. e) verə
Exercise
151 a) furl
1.0.
-1 radian –
Also 10
2.% 360
180 Thus, to convert radians to degrees, naceasary to multiply the angle by 1800 convert from degrees to radians
YC 180
180°
T
radians
5718
it is
and to multiply
To find the length of ano of a cirole radius“ sübtending an angle ✪ radians at its centra
Since the size of the angle at the centre of a circle is proportional to the arc subtending it, it follows that (5-length of arc subtending / e-
circumference of circle
member of radians subtended at the onetre by its circumference
27
To find the area of a sector subtending 6" at centre of the circle radius-r-
Let A be the area required.
Since the area of a sector is proportional to the angle it subtends at the centre, it follows that
area of circle
271
27
Example 1. Find the number of seconds in the angle subtended at the centre of a circle of radius five miles by an arc of length 2: fest.
Let ☺ be the required angle in radians,
aro - 1 8 × 5 × 1760 x 3 x @ fest
2
x 1760 x
radians
180
5x1760 degree
544 × 60 × 60
15.6 seconds
Example 2. Find the numerical value of
(a) oinoks cot
(b) (3107+008) (sin
cos) sec
(a) Exp - Bin (100) cos (100) out (100
sin cos 30° cot 45°.
~)-(1)
(b) Exp - (sin 30° + cos 30°)(sin60° – cou60°)sec60°, (cos30° + sin30°) (cos30° - sin30°) asc 60°
(cos 30 - sin
sin230) sec 60
- [(~√ √ √2 - ( } )2 ] (2)
sin
-.19 28*
-0.3333
Example 5. Express in radians the exterior and interiors of a regular polygon which has n sides.
(1) Sun of ext. /a 4 It - 27
#ach ext./, 27
Tedians
A
(ii) each int. each ext.
each int...
(1 − 2)7
rádiana
Example 6. Two angles are auch that their difference, is 200 and their sum is 14 radiana; find the ang in degrees, and also in radiana (take 7-
Let A & B be the two anglen in degres
A B
A+ B-14 r
180° 7.
(2) LA - 1 (20°
- (20 + 85 10°) = 52 2/2
270
(2)- (1)
-(85 10-20°)
-32222
In radians A
52
180
32
Exercise
SEBE
An aro of 17 yd. 1 rt. 3 in. subtends at the centre of a circle an angle of 1.9; find the radius of the circle in inches.
2. Find the value of
3tan2 + cos - cot
Bin
Show that the aum of the squares of
sine + sin( e) and cose
equal to 2
Solutions for Exercise.
1. (a) 4 coso sin 270 - Joos180 tan45°
sin270
Bin 270° = (-1)2
Exp.
- sin(180° + 90°) - aingo
4(1)(1) ~ 3(−1)(1) =
2c088090
c09360°
Frecia
Read the following passage carefully---
The peculiarity of a millionaire is that he continually risks his money in order to make more. He chases riches with the same passion that a scientifio man feels in exploring the secrets of nature, a traveller in visiting new lands, an inventor in making new machines, an artists in drawing new pictures that will surpass his former onds, a scholar in acquiring fresh knowledge, a conqueror in getting additionel territories for his empire. One millionaire will try hard to surpass another for the sake of vistory rather than of profit. Therefore in his own line he strives to be a king. Kings they are often called as the Oil king, the Silver king, and the Railway King. One- millionaire in fighting another will spend enormous
uns rather than be beaten.
Mr. Carnegie, who was the late Steel King of the United States, decided at the age of sixty that he should retire from dallar-hunting, Such an example is Tare and even then it occured very late in his life. As a rule millionaires never rest. They have bean used to excitement, and they must have something to do. The handling of an immense concern is as full of. human interest as the Secretaryship of the Colonies, and is not relying like the Secretaryship, on the popular will..
The character of a miser is somewhat differant He will not risk his money by investing it. He hoards its and for this reason denies himself the commonest comforts. He will even die or destitution rather than spend money on what is needed for his health. He will hoard his money in order to feast his eyes on the glittering heap.
How is such a character to be explained? "Mad", some will say; but this cannot be, unless all hermita are mad. A miser who dies of self-inflicted suffering is as true a hermit as any monk that ever lived, however much he may differ from a monk in motive. Further more, a miser has a great deal of method: hey never wanders, never swerves. He is honest too: ". does not rob other people: he is much too cautious" for that. But a lunatio does not stick at anything.
Miserliness is thrift carried to a wicked end.} The root-cause of thrift is, fear of future want.
That fear, if it grows, as it may do in a man of bad) temper, becomes by degrees. the master-motive of his life, and makes him at last a miser. A man never becomes a miser in a day; it takes years to make him one. He begins, with loving money because of the safety that it gives him, and ends with loving it for its own sake, The ardoar for safety the object-is transferred to the means of safety-money.
For a millionaire one can feel respect for a miser only scorn or bity. We respect the former for his talent, energy, and cleverness. Millionaires. distribute money through the community not only by their expenditure but also by their investments misers only lock it up. A millionaire is a man of enterprise. and courage; a miser is a coward. A millionaire will sometimes offer very large sums for) public purposes. It is sickening to make a hero of him for this cause, for there might be far less self- denial in the gift than when a widow give away her mite, and there might be a good deal of a lf-glorifi cution in the motive. Yet we must give him credit;- for in part with a million to endow a university, or a museum; or an observatory, he surrenders what it costs him a great deal of work to get, and what it is very
Valuab
for the public to have." Question: In not more than 220 words, W
, write a summary of the passage to show the difference between a millionaire and a miser.
(未宗轉入第六張第三頁)
(d) sin930° sin(930° 2 x 360°) Bin210°
sin30° = -0.5.
Example 3. Find the length of the chord PQ which cuts off an are 12 am long from a circle, centre 0, radius 5 om.
aro PAQ 12 cm.
Since the length of are subtending
Le is given by the formula.
aro PAC
POQ
radius
POQ = 2.4 radians
Draw ON perpendicular to PQ3 then _ PON - - __ POQ
68°45
(b) 3 sinoo secl80°
Exp. 3(0)(-1)-2(1)
0-2 - 1 w
(a) sin (−20°) × -sin20°
· 008(-20°) - 00820°
tan(-20°) = -tan20°
(b) sin(-110°)
cos(-110°). Cos110
tan(-110)
tanl10
sin(360° + 50°) - sin50° - 0.1660
cos(360° +50°) 008500 0,6428 1.1918
cos(930° - 2 x 360°) ■ cos210°
cos930
-00930°
tan930
tán 939
tan30°
- -0.866
(e) Bin(-210°). -sin210
2 x 360°) ■ tan210°)
+0.5774
- sin30°
+0.5 008(-210°) – cos210- -00830- -0.866 tan(-210°) tan210
-tan30-0.5774
-0.3420
+ 0.9397
-0.3640
Bin110
sin70
-0.9397
Cos70
-0.3420
tan 70° +2.7475
2.4 Tadians
180%
(2.4) - 137°31′
(0) sin4l0°
骂
C08410
#
tan410°
T
tan(360 + 50°) - tan50°