TRIGONOMETRY

FOR SENIOR HIGH SCHOOL X

1

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

2

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 6

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 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

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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. Ruler

2. Pencil

3. Arc degree

4. National calendar with islam month prediction calendar

5. 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

4

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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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).

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)

b

a

c α

f d

β e

p

q

ø

q q r

γ ∟

∟

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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, 90

0, and 30

0 and the second is triangle ruler which angle is 45

0 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, 30

0, 45

0,

600 and 90

0).

( zero degree, thirty degree, forty five degree, sixty degree,... ninety degree).

A

ø C B

6

8

b

∟

8

Project result

Problem

Give the explanation why for 00 and 90

0 we get value os Sine α = 0 at α = 0

0

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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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.

9

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 y-

componen as y-positif and x-component as x-negatif.

For example sin 1200 has related angle with sin (180

0 – 60

0) = sin 60

0. ( 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 (180

0-α) ( second quadrant)

α

Sine α = the length of the 𝑦−component of the triangle

𝑟𝑎𝑑𝑖𝑢𝑠

Cosine α =length of the 𝑥−component of triangle

𝑟𝑎𝑑𝑖𝑢𝑠

Tangent α = 𝑦−component of the triangle

x−component of the triangle

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quadrant III (1800+α) ( third quadrant) quadrant IV (360

0-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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WORKSHEET

Trigonometric Identity

Indicator

1. Proving formula of Pythagoras identities.

2. Proving formula of Tangent and Cotangent identities.

3. Proving formula of Cosecant and Secant identities.

4. Simplifying a trigonometric expression.

5. 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 = 𝐬𝐢𝐧𝜶

𝐜𝐨𝐬𝜶 , Cot α =

𝐜𝐨𝐬𝜶

𝐬𝐢𝐧𝜶 ( Tangent alpha equals Sine alpha// over

Cosine alpha, Cotangent alpha equals Cosine alpha// over Sine alpha).

2. Sin2 α + cos

2 α = 1 ( Sine squared alpha equals plus Cosine squared alpha

equals one).

3. 1 + tan2 α = sec

2 α ( one plus Tangent squared alpha equals Secant

squared alpha).

4. 1 + Cot2 α = csc

2 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:

Tan α = sin𝛼

cos𝛼 (Tangent alpha equals Sine alpha// overCosine alpha)

Look at the right triangle.

Sin α = .............. ( Sine alpha equals...)

Cos α =.............. ( Cosine alpha equals...)

So, sin𝛼

cos𝛼= ⋯… .. ( Sine alpha// over Cosine alpha equals...)

Prove the other formulas for your project!

α

y

x

r

∟

12

Project result

Problems

1. Given that cos α = 5

13 , sin β =

4

5 , determine the value of

1−𝑐𝑜𝑠2𝛽

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.

a. ( Tan α + 𝑐𝑜𝑠𝛼

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.

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

of x 0

0 30

0 45

0 ...

Value

of y

Value of y can be determine using substitution value of x to the f(x).

Y

1

600

-1

14

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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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 = (180

0 - α) + k. 360

0.

( 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. 360

0.

( 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).

a. Sin x = 1

2 3 ( Sine x equals a half of root three)

Answer

The equation is a ........... equation

So we can use ............ number.......... that is..............

Sin x = 1

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, 0

0 ≤ 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).

2. Find the solution set cos 2x = 1

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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WORKSHEET

Sine and Cosine Rules and Triangle Area

Indicator

1. Finding the Sine rule.

2. Finding the Cosine rule.

3. Finding the Tangent rule.

4. Using the Sine rule to solve the problem.

5. Using the Cosine rule to solve the problem.

6. 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

A. Sine rule

Consider that:

sin A = 𝐶𝐷

…=

ℎ

… ( 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

𝐒𝐢𝐧𝑨=

𝒃

𝑺𝒊𝒏 𝑩 ( a over Sine A equals b over Sine B)

In the same way results

A D

C

B

h

b a

c

∟

19

𝑎

Sin 𝐴=

𝑐

…. ( a over Sine A equals c over Sine C)

So,

𝒂

𝐒𝐢𝐧𝑨=

𝒃

𝑺𝒊𝒏 𝑩=

𝒄

𝑺𝒊𝒏 𝑪

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 = b

2 + c

2 – 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 = a

2 + c

2 – 2ac Cos B ( b squared equals a squared plus c squared

minus two ac Cosine B)

c2 = a

2 + b

2 – 2ab Cos C ( c squared equals a squared plus b squared

minus two ab Cosine C)

C. Triangle areas

See ∆ABC ( triangle ABC)

Area of the triangle is L = 1

2 c.h ( L equals a half of c times h)

See ∆ADC ( triangle ADC)

Sin A = ℎ

…↔ ℎ = ⋯ ( Sine A equals h over... so h equals...)

Thus, area of ∆ABC is L = 1

2 c.h ( triangle ABC,... L equals a half of c

times h)

= ....

With the same way, prove the formula of triangle areas L = 1

2 𝑎𝑏 sin 𝐶 and

L= 1

2𝑎𝑐 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 𝐬𝐢𝐧𝑩 𝐬𝐢𝐧 𝑪

𝟐 𝐬𝐢𝐧𝑨 ; 𝑳 =

𝒃𝟐 𝐬𝐢𝐧𝑨 𝐬𝐢𝐧 𝑪

𝟐 𝐬𝐢𝐧𝑩 ; 𝑳 =

𝒄𝟐 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧𝑩

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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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 right-

angled 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.