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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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