Download TRIGONOMETRY

TRIGONOMETRY
FOR SENIOR HIGH SCHOOL X
PREFACE
Considering the numbers of the international school in Indonesia, the writer
concerned to provide a worksheet that can help teachers or student teachers that
will hold their mathematic class using English.
This worksheet includes direction to hold learning in the real condition, and
possibly make coopreative learning. This worksheet is writen based on the project
based learning that expect to make the students be more responsible, directed, and
accustomed to solve problems.
Mathematical expressions that are read aloud often makes the student
teachers confused how to read them. This book is added a guideline to read the
formula and mathematical expressions, for example:
1. Each mathematical expression in this worksheet is accompanied by direction
to read, the printed in Italic font style. Example tan x = 300 read as Tangent x
equals thirty degree.
Direction
Sign “ ,... “ means word(s) that is (are) not mathematical expression.
Sign “//” means caesura.
2. This direction is used to guide the teacher and students in reading
mathematical expressions.
Further more, this book will become a guideline for people of Indonesia
especially teachers and students of billingual school in reading ( aloud) the
mathematical formulas.
writer
1
CONTENT
Angle and Measurement
2
A. Degree definition
2
B. Angle measurement 3
Trigonometric Ratios
6
A. Trigonometric ratio formed by An Angle of A Right Triangle
B. Value of Trigonometric Ratio for special angle
7
C. Trigonometric formulas of related angles
8
Trigonometric identity
11
Trigonometric Function
13
Simple Trigonometric Equation
6
16
Sine and Cosine Rules and Triangle Area
18
A. Sine Rule
18
B. Cosine Rule 19
C. Triangle Area 19
Glossary
21
Reference
21
2
WORKSHEET
Angle and Measurement
Indicator:




Understanding definition of angle.
Identifying negative angle and positive angle based on initiate side
and terminal side.
Converting degree-minute-second form to decimal degree form if a
degree-minute-second form is given and vice versa.
Converting angle unit: degree to radian, radian to degree.
Equipments
1.
2.
3.
4.
5.
Ruler
Pencil
Arc degree
National calendar with islam month prediction calendar
Compass
A. Degree definition
Follow the instruction below to finally define the degree definition!
Direction:
1. Draw two lines named OA and MC
2. Use the compass to draw arc AP that is rotate line OA counter clockwise
by centre point O then draw line OP.
3. Use the compass to draw arc CQ that is rotate clockwise line MC
clockwise by centre point M then draw line MQ.
Project result
Questions
3
1. Mention the angles resulted from your drawing!
2. Which one is the positive angle and which one is the negative angle?
3. Define the components the angles? ( Arms and initiate side and terminal
side of the angles).
Conclussion
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B. Angle Measurement
Follow the instruction below to finally be able to convert degree units!
Direction
1. Take attention to the islam month prediction in the calendar!
2. Note all of the angle size in the month prediction, example 170 3’ 12”.
( seventeen degree three minutes and twelve seconds).
Project result
Problems
1. Express those angles in the form of decimal degree
2. If α = 300 34’ 20”, β = 210 40’ 12”, find 2α – β in the decimal degree
form. ( alfa equals thirty degree thirty four minutes and twenty
seconds).
3. Another way to express the degree size that is used radian, express the
angle from the problem 1 to the form of radian.
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Conclussion
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WORKSHEET
Trigonometric Ratios
Indicator
1. Determining Sine, Cosine, and Tangent of angle based on ratio of right
triangle.
2. Determining Sine, Cosine, and Tangent from special angle.
Equipments
One set of triangle ruler
Text book
A. Trigonometric ratio formed by An Angle of A Right Triangle
Follow the instruction below to finally determine the trigonometric Ratio!
Direction
Observe the following triangles, and choose the triangle that can form
trigonometric ratio and define the ratio ( Sine, Cosine and Tangent).
q
∟
a
r
q
q
f
d
p
c
b
γ
α
β
ø
∟
e
Project result
Problems
1. There are other trigonometric ratio which are the reverse of Sine,
Cosine and Tangent, they are respectively called Secant, Cosec and
Cotangent. Define them from the project result!.
2. Determine the value of Sine, Cosine and Tangent of ø from following
triangle. ( teta)
6
A
b
6
∟
B
ø
8
C
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Conclussion
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B. Value of Trigonometric Ratio for special angle
Follow the instruction below to finally determine value of Trigonometric
Ratio for special angle!
Direction
Pay attention to your set of triangle ruler.
There are two types of the right triangle, the first is triangle ruler which angle
is 600, 900, and 300 and the second is triangle ruler which angle is 450 and
900. ( sixty degree, ninety degree, thirty degree,... forty degree and ninety
degree).
Measure the length of each of the ruler side and than use the trigonometric
ratio to find the value of trigonometric ratio for special angle ( 00, 300, 450,
600 and 900).
( zero degree, thirty degree, forty five degree, sixty degree,... ninety degree).
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Project result
Problem
Give the explanation why for 00 and 900 we get value os Sine α = 0 at α = 00
and Sine α = 1 at α = 900. ( zero degree, ninety degree,... Sine value equals
zero at zero degree and Sine value equals one at ninety degree).
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Conclussion
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C. Trigonometric formulas of related angles
Follow the instruction below to finally determine the Trigonometric formulas
of related angles!
Direction
To determine value of the obtuse angle we need to determine the related
angles. Before that, we should understand the definition of trigonometric ratio
in any quadrant.
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Sine α =
α
the length of the 𝑦−component of the triangle
Cosine α =
𝑟𝑎𝑑𝑖𝑢𝑠
length of the 𝑥−component of triangle
𝑟𝑎𝑑𝑖𝑢𝑠
𝑦 −component of the triangle
Tangent α = x−component
of the triangle
For example, trigonometric ratio in the second quadrant, the right triangle is
constructed from x negatif, y positif and angle (1800- α). So we define ycomponen as y-positif and x-component as x-negatif.
For example sin 1200 has related angle with sin (1800 – 600) = sin 600. ( Sine
one hundred and twenty degree,... Sine// one hundred and sixty minus sixty
degree// equal to Sine sixty degree).
Pay attention to your text book, draw the trigonometric ratio for related angle
for all quadrant then determine the value of the trigonometry ratio.
Project result
quadrant I (900-α) ( first quadrant)
quadran II (1800-α) ( second quadrant)
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quadrant III (1800+α) ( third quadrant)
quadrant IV (3600-α) ( fourth quadrant)
Problem
Express trigonometric form below in acute angle!
a. Sin 2410 ( Sine two hundred and forty one degree).
b. Tan 1750 ( Tangent one hundred and seventy five).
c. Cot 3310 ( Cotangent three hundred and thirty one).)
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Conclussion
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WORKSHEET
Trigonometric Identity
Indicator
1.
2.
3.
4.
5.
Proving formula of Pythagoras identities.
Proving formula of Tangent and Cotangent identities.
Proving formula of Cosecant and Secant identities.
Simplifying a trigonometric expression.
Identifying trigonometric identities expression base on 4 basic
trigonometric identities.
Direction
Follow the instruction below to finally prove, simplify and identify trigonometric
expression!
Trigonometric identity is an open sentence, you can prove that through
changing the left hand side expression into the same pattern of expression with
the right hand side expression, or changing the right hand side into the same
pattern of expression with the left hand side expression.
And the basic trigonometric identities that you can use are:
1. Tan α =
𝐬𝐢𝐧 𝜶
𝐜𝐨𝐬 𝜶
, Cot α =
𝐜𝐨𝐬 𝜶
𝐬𝐢𝐧 𝜶
( Tangent alpha equals Sine alpha// over
Cosine alpha, Cotangent alpha equals Cosine alpha// over Sine alpha).
2. Sin2 α + cos2 α = 1 ( Sine squared alpha equals plus Cosine squared alpha
equals one).
3. 1 + tan2 α = sec2 α ( one plus Tangent squared alpha equals Secant
squared alpha).
4. 1 + Cot2 α = csc2 α ( one plus Cotangent squared alpha equals Cosec
squared alpha).
To prove the base of trigonometric identities, you can use the Phytagorean
property.
The Proof:
sin 𝛼
Tan α = cos 𝛼 (Tangent alpha equals Sine alpha// overCosine alpha)
Look at the right triangle.
Sin α = .............. ( Sine alpha equals...)
Cos α =.............. ( Cosine alpha equals...)
So,
sin 𝛼
cos 𝛼
r
y
∟
= ⋯ … .. ( Sine alpha// over Cosine alpha equals...)
α
x
Prove the other formulas for your project!
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Project result
Problems
1. Given that cos α =
5
13
, sin β =
4
5
, determine the value of
1−𝑐𝑜𝑠 2 𝛽
𝑠𝑖𝑛 ∝.𝑐𝑜𝑠 ∝
, use the
trigonometric identities! ( Cosine alpha equals five over thirteen, Sine beta
equals four over five,...one minus Cosine squared beta// over Sine alpha
time Cosine alpha).
2. Solve then following identities.
𝑐𝑜𝑠𝛼
a. ( Tan α + 1+𝑠𝑖𝑛 ∝ ) sin α = tan α. ( Tangent alpha plus// Cosine alpha//
over one plus Sine alpha).
b.
𝑐𝑜𝑠𝛼
=
1+𝑠𝑖𝑛 ∝
1−𝑠𝑖𝑛𝛼
𝑐0𝑠∝
( Cosine alpha// over one plus Sine alpha// equals one
minus Sine alpha// over Cosine alpha).
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Conclussion
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WORKSHEET
Trigonometric Functions
Indicator
1. Explaining the concept of Sine function using circle
2. Explaining concept of Cosine function ussing circle
3. Explaining concept of Tangent function using circle
Direction
Follow the instruction below to finally explain the concept of Trigonometric
Function!
Trigonometric function is a function that maps each member of a real number
to a member of a real number member of relation of a trignometric ratio.
𝑓 ∶ 𝑥 → 𝑓 𝑥 , with 𝑥 ∈ 𝑅 is in angle size of degree or radian. ( function f
maps x to f(x)).
The procedure to draw a trigonometric function are as follows.
a. Express the Cartesian field, X-axis represented size angle, Y-axis is the
function value of x.
Y
1
600
-1
Note: to make a scale you an use π = 3.14, Hence 360 is 2π is equal to 6.28
unit scale. In the easy way, scale 1 unit equals to 600.
b. take the special angles in the X-axis then determine the corresponding
value of the function in the f(x). ( f of x or f(x)).
Value
...
00
300
450
of x
Value
of y
Value of y can be determine using substitution value of x to the f(x).
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c. if it needed, make some scale of the X-axis and Y-axis.
d. Connect the results point in (b), then draw a sincere curva.
Graph for y= sin x and y = cos x.
Plot the functions below!
a. f(x)= sin x, f(x)= sin 2x in one cartesian field. ( f(x) equals Sine x, f(x)
equals Sine// two x).
b. f(x)= cos x, f(x)= 2 cos x in one cartisian field. ( f(x) equals Cosine x, f(x)
equals two// Cosine x).
c. f(x)= tan x. ( f(x) equals Tangent x).
Project result
Problems
1. Find the differences between F(x)= sin x and f(x)= sin 2x. ( f(x) equals
Sine x, f(x) equals Sine// two x)
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2. Describe the curve of f(x)= cos 5x based on your discovery, without
plotting. (f(x) equals Sine// five x).
3. Why is the curve of f(x)= tan x discontinues in some points? ( f(x) equals
tanget x).
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Conclussion
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WORKSHEET
Simple Trigonometric Equation
Indicator
1. Finding the solution set of Sine equation.
2. Finding the solution set of Cosine equation.
3. Finding the solution set of tanget equation.
Direction
Follow the instruction below to finally solve the Trigonometric Equation!
Trigonometric equation has trigonometric ratio in it. Find the angles that
satisfy the equation which are called by solution set.
Remember that
The following trigonometric equation formulas
a. Sin x = sin α
x = α + k. 3600 or x = (1800 - α) + k. 3600.
( Sine x equals Sine alpha)
( x equals alpha plus k times three hundreds and sixty degree,... x equals
one hundred and eighty degree minus alpha// plus k times three hundreds
and sixty degree).
b. Cos x = cos α
x = α + k. 3600 or x = -α + k. 3600.
( Cosine x equals Cosine alpha)
( x equals alpha// plus k times three hundreds and sixty degree,... x equals
minus alpha plus k times three hundreds and sixty degree).
c. Tan x = tan α
x = α + k. 1800
( Tangent x equals Tangent alpha )
( x equals alpha plus k times one hundred and eighty degree).
Fill in the blanks!
Determine the solution set in interval 0 ≤ 𝑥 ≤ 2𝜋. ( x is more than or equal to
zero and x is less than or equal to two x).
1
a. Sin x = 2 3 ( Sine x equals a half of root three)
Answer
The equation is a ........... equation
So we can use ............ number.......... that is..............
1
Sin x = 2 3 ( Sine x equals a half of root three)
= sin ... ( Sine...)
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≫ 𝑥 = ⋯ + 𝑘. 3600
or
≫ 𝑥 = 1800 − ⋯ + 𝑘. 3600
( x equals... plus k times three hundreds and sixty degree or x equals one
hundred and eighty degree minus...// plus k times three hundreds and sixty
degree).
For k = 0 → 𝑥 = ⋯
or for k = 0 → 𝑥 = ⋯
( k equals zero then x equals... , k equals zero then x equals... )
So the solution set ={..., ....}
Project result
Problems
1. Find the solution set for cos x =
1
2
3, 00 ≤ 𝑥 ≤ 3600. ( Cosine x equals a
half of root three for x more than or equals zero degree and x less than or
equal to three hundreds and sixty degree).
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2. Find the solution set cos 2x = 2, 0 ≤ 𝑥 ≤ 2𝜋. ( Cosine two x equals a half
for x more than or equals zero degree and x less than or equal to two Phi).
3. Find the solution set sec 3x - 2 = 0, for 0 ≤ 𝑥 ≤ 2𝜋. ( Secant three x
minus root two equals zero for x more than or equals zero and x less than
or equal to two Phi).
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Conclussion
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WORKSHEET
Sine and Cosine Rules and Triangle Area
Indicator
1.
2.
3.
4.
5.
6.
Finding the Sine rule.
Finding the Cosine rule.
Finding the Tangent rule.
Using the Sine rule to solve the problem.
Using the Cosine rule to solve the problem.
Using the Tangent rule to solve the problem.
Direction
Follow the instruction below to find The Trigonometric Rules and finally solve
problem using them!
The Proof of the Sine and Cosine Rule
To prove the Sine and Cosine rule, use this triangle
C
A. Sine rule
b
a
h
A
Consider that:
sin A =
𝐶𝐷
…
ℎ
=…
∟
D
B
c
( Sine A equals CD over ... equals h over...)
h = ..... Sin A .......(1) ( h equals ... times Sine A as equation one)
sin B =
𝐶𝐷
…
=
ℎ
…
( Sine B equals CD over... equals h over...)
h = ....Sin B ........(2) ( h equals ... times Sine B as equation two)
From equations (1) and (2), give
...... = ......, .................(3) ( as equation three)
According to the equation (3) results
𝒂
𝐒𝐢𝐧 𝑨
𝒃
= 𝑺𝒊𝒏 𝑩 ( a over Sine A equals b over Sine B)
In the same way results
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𝑎
𝑐
Sin 𝐴
= ….
( a over Sine A equals c over Sine C)
So,
𝒂
𝒃
𝒄
= 𝑺𝒊𝒏 𝑩 = 𝑺𝒊𝒏 𝑪
𝐒𝐢𝐧 𝑨
B. Cosine Rule
Consider triangle ADC
Cos A =
𝐴𝐷
…
or AD = ....Cos A ( Cosine A equals AD over... or AD
equals... times Cosine A)
(DC)2 = (AC)2 – (...)2 ( DC// squared equals AC// squared minus ...//
squared).
= b2 - ......
( b squared minus....)
Triangle BDC satisfies
(BC)2 = (BA -...)2+ (DC)2 ( BC// squared equals BA minus ...// squared
plus DC// squared).
= (c - ......)2 + ......... ( c minus ...// squared plus...)
↔ a2 = b2 + c2 – 2bc Cos A ( so we get a squared equals b squared plus c
squared minus two bc Cosine A)
Use the same concept to get
b2 = a2 + c2 – 2ac Cos B ( b squared equals a squared plus c squared
minus two ac Cosine B)
c2 = a2 + b2 – 2ab Cos C ( c squared equals a squared plus b squared
minus two ab Cosine C)
C. Triangle areas
See ∆ABC ( triangle ABC)
1
Area of the triangle is L = 2 c.h ( L equals a half of c times h)
See ∆ADC ( triangle ADC)
ℎ
Sin A = … ↔ ℎ = ⋯ ( Sine A equals h over... so h equals...)
1
Thus, area of ∆ABC is L = c.h ( triangle ABC,... L equals a half of c
2
times h)
= ....
1
With the same way, prove the formula of triangle areas L = 2 𝑎𝑏 sin 𝐶 and
1
L= 2 𝑎𝑐 sin 𝐵. ( project 1). (L equals a half of ab Sine C,... L equals a half
of ac Sine B).
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Information

If given a side and two angles, the formula to find the triangle area are as
follows.
𝑳=

𝒂𝟐 𝐬𝐢𝐧 𝑩 𝐬𝐢𝐧 𝑪
𝟐 𝐬𝐢𝐧 𝑨
;𝑳=
𝒃𝟐 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑪
𝟐 𝐬𝐢𝐧 𝑩
;𝑳=
𝒄𝟐 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩
𝟐 𝐬𝐢𝐧 𝑪
( L equals a squared Sine B Sine C// over 2 Sine A, L equals b squared
Sine A Sine C// over 2 Sine B, L equals c squared Sine A Sine B// over 2
Sine C).
If given of all sides, the formula to find the area of triangle are as follows.
𝑳=
𝒔 𝒔 − 𝒂 𝒔 − 𝒃 (𝒔 − 𝒄) with 𝒔 =
𝒂+𝒃+𝒄
𝟐
( L equals root// s times// s minus a// s minus b// s minus c,... s equals a
plus b plus// over 2).
Project result
Problems
1. Calculate area of the triangle ABC given, if the dimension known as
follows!
a. a = 9 cm, b = 12 cm and c = 15 cm
b. a = 20 cm, b = 15 cm and C = 300
2. A parallelogram has a and b length units diagonals. If the angles between
these diagonal is α, show the area of the parallelogram of
1
2
𝑎𝑏 sin 𝛼! ( a
half of ab sine alpha).
3. Can you find the triangle area if you are given two angles and one side?
Explain your answer!
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Conclussion
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Glossary
Angle
: the figure formed by two line segments that extend
from a common point, or by regions of two planes that
extend from a common line.
Degree
: a measure of angle equal to one 360th part of the angle
traced out by one full revolution of a line segment
around one of its endpoints, written °.
Right Triangle
: a triangle in which one of the angles is a right angle
Trigonometric Function : n. any of a group of functions expressible naively in
terms of the ratios of the sides of a right-angled triangle
containing an angle equal to the argument of the
function in radians, or, more generally for real
arguments, in terms of the ratios of the coordinates of
the points on the circumference of a circle centered on
the origin and the radius as the latter sweeps out that
angle, as shown in the figure below for an angle in the
second quadrant.
Sine
: n. the trigonometric function that is equal in a rightangled triangle to the ratio of the side opposite the
given angle to the hypotenuse.
Cosine
: n. a trigonometric function that in a right-angled
triangle is equal to the ratio of the side adjacent the
given angle to the hypotenuse.
Tangent
: abbreviated tan. a. a trigonometric function that in a
right-angled triangle is the ratio of the length of the side
opposite the given angle to that of the adjacent side,
where the lengths are taken to be positive, that is, y/x.
Reference
BNSP. 2006. Model silabus mata pelajaran SMA/ MA. Jakarta: BP. Cipta
Jaya.
Marwanto dkk. 2008. Mathematics. Jakarta: Yudistira.
Mathematics Forum. 2006. Mathematics for Senior High School Year X.
Yogyakarta: Yudhistira.
Morrison, Karen. 2002. IGCSE. Mathematics. Cambridge: Cambridge
University Press.
PASIAD. 2010. Lise Annual Plan. Islamabad.
_______. 2010. Trigonometry modul of zambak.
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