TRIGONOMETRI
TRIGONOMETRI
TRIGONOMETRI
What is TRIGONOMETRY??
Trigonometry is the most well-known branch of
mathematics. We study about angle, triangle, sine,
cosine, tangent, etc.
By using trigonometry, an astronomer is able to
estimate the distance between the earth and the moon,
the diameter of planets, etc.
In the end of this chapter, you are expected to
have understood and use simple trigonometry in
daily life. By studying and understanding the
fundamentals of trigonometry, you can at least
calculate the height of things such as buildings,
towers or trees by using number of simple facts
about angles and triangles.
Example:
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Two teachers have same
height. They are seeing the
top of the flagpole.
Dua orang guru dengan tinggi badan
yang sama sedang berdiri
memandang puncak tiang bendera di
sekolahnya.
The first teacher is 10 m in
front of the second teacher.
Guru pertama berdiri tepat 10 meter
Could you calculate the
dari guru kedua. length of the flagpole?
Dapatkah kamu menghitung tinggi
the first teacher’s elevation tiang bendera tersebut??
angle is 60o and the second HOW TO DETERMINE THE
teacher is 30o LENGTH OF THE
Jika sudut elevasi guru pertama 60o FLAGPOLE?
dan guru kedua 30o. bagaimana cara menghitung
tingggi tiang bendera??
When surveying a swamp, a surveyor walked 425
meters from point A to point B, then turned 65
degrees and walked 300 meters to point C.
calculate the length of the AC.
Pada saat mensurvei sebidang rawa-rawa, seorang pensurvei
berjalan sejauh 425 meter dari titik A ke titik B, kemudian
berputar 65 derajat dan berjalan sejauh 300 meter ke titik C.
hitung panjang AC.
A firefighter wants to reach a window which its length
is 4 m, by using a stairs and considering safety factors,
the minimum distance between stairs and wall is not
less than 1 meter, how is the length of the stairs that
can be used? and how to locate the stairs?
Seorang pemadam kebakaran ingin meraih jendela yang
tingginya 4 m, dengan menggunakan tangga dengan
memperhatikan faktor keselamatan, jarak minimal yang
dibentuk tangga dengan dinding tidak kurang dari 1 meter,
berapa tinggi tangga yang dapat digunakan ? bagaimanakah
penempatan tangga tersebut ?
In a village which is far from the airport, children have a habbit
of following the airplane that crossing their village. Bolang
observe the airplane that flies 120 Km. Bolang elevation angle
towards the airplane is 30 degree. Determine the distance
between Bolang and the airplane.
Unit Degree
Definition of Degree:
1. When the circumference of a circle is devided into
360 parts, the central angel corresponding to one of
these arcs is called 1 degree denoted by 1o
2. Order to measure smaller angles we use smaller
angle units. Each degree can devided into 60 equal
parts whisch is called by minutes. 1 minutes is
denoted by 1’.
3. Each minutes can be divided into 60 equal parts
which is called by second. 1 minutes is denoted
by 1”.
QUESTIONS
1. 1 FULL rotation = … o (degree)
1
2. 2 rotation = … o (degree)
1
3. rotation = … o (degree)
4
1
4. rotation = … o (degree)
360
Example
Consider an angle which measures 37
degrees, 45 minutes, and 30 seconds. We
can write this angle in two ways:
In degree-minute-second form: 37o 45’ 30”
In decimal degree form: 37.7583 o
Exercise:
1. Write 56o 20’15” in decimal degree form
2. Write 17.86o in degree-minute-second form
RADIAN
Let AB be an arc of a circle with radius r
such that the arc length of AB is also r.
Then the measure of the central angle
corresponding to AB is called 1 radian. It is
denoted by 1 rad or 1R
Fill the blank!
Given arc AB = 1 radian
𝐜𝐢𝐫𝐜𝐮𝐦𝐟𝐞𝐫𝐞𝐧𝐜𝐞 𝐨𝐟 𝐜𝐢𝐫𝐜𝐥𝐞
1 full rotation = radian
𝐚𝐫𝐜 𝐀𝐁
…
= radian
𝐫
= … radian
CONCLUSION
𝝅
1= rad
𝟏𝟖𝟎
180o
1 rad = = 57,3 o
π
π = 180 o
Commonly used angles in radians
Degrees Radians Degrees Radians
0o 0 180o …(4)
30o π …(5) 7π
6 6
45o π 210o …(6)
4
60o π 225o …(7)
3
90o π …(8) 3π
2 2
…(1) 2π …(9) 5π
3 3
…(2) 3π …(10) 7π
4 4
…(3) 5π 330o …(11)
6
JUST WHAT YOU NEED
Right-angled Triangles
Segitiga siku-siku
𝒄
𝒂
Characteristics:
One angle is at 90 o
If 𝑐 is the hypotenuse, 𝑎 and 𝑏 𝒃
are right angled sides, then
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
(Pythagoras’ Rule) always
holds.
TRIGONOMETRIC RATIOS IN
RIGHT-ANGLED TRIANGLE
ON A RIGHT-ANGLED TRIANGLE
The sine of an angle is the ratio
Opposite
ON A RIGHT-ANGLED TRIANGLE
The cosecant of an angle is the
ratio between the side
Opposite