1969-11-11 — Page 22

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數學科 (二)

MATHEMATICS (2)

Circular Functions of the Acute Angle. If ABC ba

triangle right-angled at A, with the acute angle

ACB e, then the following ratios are known as the circular or trigonometric functiones

AB

• СА

BC

DE

CA

and are denoted by sin e, cosine 8, tangent e, cosecant 9, secant 0, and cotangent 6, or more briefly sin 9, cos 8,tan 9, coseo 8, sec @ and oot → respectively.

From the definitions

報日僑華

二期星日一十月一十年九六九一曆公年八十五國民華中 青教僑華

tano

an: 0

(ii) The angles 30° and 60. Consider the equilateral

triangle ABC of aide 'a', with AD drawn perpendicular to BC.

。育教僑

僑華

* 09800

tan

cos e

COS

cot

COB:9 sin:0

Alao, since BOAB and BC:

AC

and since AB + CA

sin 0. 1, cos 041, COB

BC,

22, and sec

+ ≥2

совбо

3. Es cot of.

find the value

q sind

By geometry D is the mid-point of 50 and AD bisects BAC

BAD

30°

Prcos L

Þ cooα + q sin A

Find a solution for each of the followin equations:

(i) 8 sin

+6 COM O

(ii) 2 sec 6 x 1 1 3 tan

Exercise Z

Solution

Since sin @=

m

m

From Pythagoras' theoTEM

in300

1.e. sin + cos

Using Pythagoras' theorem on the above triangle,

AB +CA BC

Dividing through (1) by

AB

BO

(2)

-//] (tan30° - - -

(iii) The angle 45°. Consider the isosceles triangle

ABO right-angled at A with AB AC

When LABO

Br. Fythagoras BC

459

AB+

BC = W/28

Dividing through-(a) by

1+

Dividing through (2) by sin

sine

cotˆ0 + 1 # cose

(4)

ऋ

tan45

(iv) The angle 90° Using the diagram and notation

of (1), ár 9 approaches zero / CBA approaches .900

that (a)

tan α- cot cot d tan (b) (taná+RecA)2

tan d

cot a

cot

ĐẢO BA

@ind

COB CBA-

Tan CBAN

CA AB

sin90°

A

BC

tango"

tan

sind

The complete set of results is given in the following table.

GOB X

Angle

sin

Sine

0.0

300

45°

sin A

.COBA

sia X sin 5 -

cos 3). sản ở sin

Comine.

No3

008 in S. (COBα COB

Bin sin

~~~

900

hoa bín

Tangent

tan doo

cot B

cottan

cot A

by Pythagoras theorm, the adj. side

b= (m2+1)* −(m = 1)~

tane=

not

Solution

ely stick held vertically

length of stick

length of shadow

AC

듣.

cof e

AC

BC Cote

3 cet 52°

3 × 0.7813

2:3439

Ans. The length of shadow

2.34 ft.

cll) To have the longest shadow,

SUN

(tan A+ Bec A (ziná + 1)2

COBA

(#ind+1)2

1 - sin A

1- Bina

(tana + seÇA)

aini sina

Trigonometric Functions of the Angles 0o.

The most important trigonometric functions are sins, cosine, and tangents and these will be the only ones considered in this case, since those for the other three trigonometrio functions can be obtained from the by inverting the reaulte.

(1) The angle 0° - Consider the triangle ABC right-

angled at A with / BCA -.9, where ♦ is very anall.

As

approaches zero, A, B tend to coincide, 1.6. as approaches zero, IB approaches zero, and CB tends to equality with CA.

sino°

BG

COMO

52°

the stick is placed abliquely

Example 2 Solve 3 tan.

30

mine

+ cot

cas sine

Собес

sine

So

+ COB 9-5. Coae

3(1

-COB

cos28) + cos28 - 5 cont

20082 + 5 cos → → 3 - 0

(2 cos 0-1) (cos

3) - 0

61 8

or −3 (rejected).

Exercise 2

Práve the following identities :

(a) (tan -

*)2

- 2cosece (oote

2806.

(b) (1 + sinė + cone)" - 2(1 + sine) (1 + cos

If x - segð – tanë, prove that x + - - 2000 8, and expre

in terms of e

as to be at rt. < to the sin beam.

AB is the length of shadew

AB

cse 52°

BC csc52°

= 3 × 1·269 = 3.807

Ans. The shadow becomes

3.81 feet.

Solution

cos 15 cse 75° cos 15°• sec 15°

cos 15°.

cosis

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