8. BASIC CONSEPT OF ANGLE
Mr. Yahya was a guard of the school. The Height of Mr. Yahya is
1,6 m. He has a son, his name is Dani. Dani still class II
elementary school. His body height is 1, 2 m. Dani is a good
boy and likes to ask. He once asked his father about the height
of the flagpole on the field. His father replied with a smile, 8
m. One afternoon, when he accompanied his father cleared the
weeds in the field, Dani see shadows any objects on the
ground. He takes the gauge and measure the length of his
father shadow and the length of flagpoleβs shadow are 6,4 m
and 32 m. But he couldnβt measure the length of his own
because his shadow follow ing his progression.
PROBLEM
9. A
B E
G C
F
D
XO
Where :
AB = The height of flagpole (8 m)
BC = The lenght of the poleβs shadow
DE = The height of Mr. Yahya
EC = The length of Mr. Yahyaβs Shadow
FG = The height of Dani
GC = The Lenght of Daniβs shadow
6,4
8
1,6
1,2
32 f
10. CE
D
A
B C CG
F
g8
32
1,6
6,4
1,2
1088 43,52
f
πΉπΊ
π·πΈ
=
πΊπΆ
πΈπΆ
=
1,2
1,6
=
π
6,4
. f = 4,8
πΉπΆ = π = 24,48
a. ____ = ____ = ____ = ________ = ________ = ______ = ____________________ =
24,48 43,52 1088
Opposite side the angleFG
GC
DE
EC EC
AB 1,2 1,6 8
Hytenuse of triangles
0,24
the sine of the angle C,
written sin x0 = 0.24
b. ____ = ____ = ____ = ________ = ________ = ______ = _______________________ =
24,48 43,52 1088
adjacentGC
FC
EC
DC AC
BC 4,8 6,4 32
Hypotenuse of triangle
0,97
the cosine of the angle C,
written cos x0 = 0.97
c. ____ = ____ = ____ = ________ = ________ = ______ = _______________________ =
4,8 6,4 32
Opposite side the angleFG
GC
DE
EC BC
AB 1,2 1,6 8
adjacent
0,25
the tangent of the angle C,
written tan x0 = 0.25
11. PROBLEM
1,5 m
8 m
9,5m
πΌ
Undu standing 8 m in front of the
pine tree with height of 9.5 m. If the
height of Undu is 1,5 m. Determine
the trigonometric ratio of Angle πΌ.
12. Where :
AC = The height Of Pine Tree
ED = The height of Undu
DC = The distance between Tree and Undu
1,5 m
8 m
A
B
CD
E πΆ
9,5 m
SOLUTION
π ππ πΌ? πππ πΌ? π‘ππ πΌ?
Find EA!
8 2
πΈπ΄ = πΈπ΅2 + π΄π΅2
= 82 + 9,5 β 1,5 2
= 64 + 64
= 128
= 8 2
πππ πΌ =
8
8 2
=
1
2
2
π‘ππ πΌ =
8
8
= 1
π ππ πΌ =
8
8 2
=
1
2
2
14. the sine of an angle is the length of
the opposite side divided by the
length of the hypotenuse.
DEFINITION
B
P J
sin π½ =
ππ΅
π΅π½
the cosine of an angle is the length of
the adjacent side divided by the length
of the hypotenuse.
πππ π½ =
ππ½
π΅π½
the tangent of an angle is the
length of the opposite side
divided by the length of the
adjacent side.
π‘ππ π½ =
ππ΅
ππ½
15. the cosecant of an angle is the
length of the hypotenuse divided by
the length of the opposite side.
Written :
DEFINITION
B
P J
cosππ π½ =
π΅π½
ππ΅
the secant of an angle is the length of
the hypotenuse divided by the length of
the adjacent side.Written:
π ππ π½ =
π΅π½
ππ½
the tangent of an angle is
the length of the adjacent
side divided by the length of
the opposite side. written :
πππ‘ π½ =
ππ½
ππ΅
cosππ π½ =
1
sin π½
π ππ π½ =
1
cos π½
πππ‘ π½ =
1
tan π½
16. S O H C A H T O A
REMEMBER
i
n
p
p
o
s
i
t
e
y
p
o
t
e
n
u
s
e
o
s
d
j
a
c
e
n
t
y
p
o
t
e
n
u
s
e
a
n
p
p
s
o
s
i
t
e
d
j
a
c
e
n
t
17. EXAMPLE
Given right triangle ABC, right-angled at β ABC. If the length
of the side AB = 3 units, BC = 4 units. Determine sin A, cos A,
and tan A.
C
BA 3 units
4 units
19. Ratio for Specific Angles
A(x,y)
x
y
r
Y
O
X
Suppose point A (x, y), the length
OA = r and the angle AOX = Ξ±.
πππ Ξ± =
Cos πΌ =
Tan πΌ =
π¦
π
π₯
π
π¦
π₯
πΌ
A(-x,y)
-x
y
r
Y
O
X
πππ Ξ± =
Cos πΌ =
Tan πΌ =
π¦
π
β
π₯
π
β
π¦
π₯
Quadrant II (90o-180o)Quadrant III (180o-270o)
Y
O
X
A(-x,-y) -x
-y
r
πππ Ξ± =
Cos πΌ =
Tan πΌ =
β
π¦
π
β
π₯
π
π¦
π₯
O
A(x,-y)
x
-y
r
Y
X
Quadrant IV (270o-360o)
πππ πΌ =
Cos πΌ =
Tan πΌ =
β
π¦
π
π₯
π
β
π¦
π₯
21. EXAMPLE
Suppose given points A(-12,5) and β XOA = Ξ±.
Determine the value of sin Ξ±, cos Ξ± and tan Ξ±
SOLUTION
x = -12 and y = 5. Quadrant II
A(-12,5)
5
O
Y
X
Ξ±
Cos π΄ = β
12
13
Tan π΄ = β
5
12
πππ π΄ =
5
13
12
ππ = 12 2 + 52
= 144 + 25
= 169
= 13
13
22. Trigonometric Ration For Special Angles
0o, 30Β°, 45Β°,60Β° and 90o
45o
45o
30o
60o 60o
M
K LP
A
B C
22
1 1
25. P(x,y)
1
1NO x
y
X
Y
αΆΏ
sin π =
π¦
1
= π¦ cos π =
π₯
1
= π₯ tan π =
π¦
π₯
If π = 0 π, then P(1,0)
β’ sin 0Β° = y = 0
β’ cos 0Β° = x = 1
β’ tan 0Β° = y/x = 0/1=0
β’ sin 90Β° = y = 1
β’ cos 90Β° = x = 0
β’ tan 90Β° =y/x =1/0, undefine
If π = 90 π
, then P(0,1)
26. Trigonometric ratios of Special Angles
πΌ 0 π 30 π 45 π 60 π 90 π
Sin πΌ 0
1
2
1
2
2
1
2
3 1
Cos πΌ 1
1
2
3 1
2
2
1
2
0
Tan πΌ 0
1
3
3 1 3 β
27. Anzar want to determine Angle size from a
trigonometric ratio. Given to her ratio as follows.
sin πΌ =
1
2
, He must to determine the value of Ξ±
(Angle size)
PROBLEM