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3Material AICLE. 4º de ESO: Trigonometry
Identificación del material AICLE
CONSEJERÍA DE EDUCACIÓN
Dirección General de Participación e Innovación Educativa
TrigonometryTÍTULO
A2.2NIVEL LINGÜÍSTICOSEGÚN MCER
InglésIDIOMA
MatemáticasÁREA / MATERIA
GeometríaNÚCLEO TEMÁTICO
4º de Educación Secundaria. Matemáticas BCORRESPONDENCIA CURRICULAR
6 sesionesTEMPORALIZACIÓN APROXIMADA
- Distinción de las razones trigonométricas de un ángulo agudo: seno, coseno y 
tangente y sus inversas, y cálculo de las razones a partir de los datos en distintos 
contextos
- Utilización de la calculadora para hallar el seno, el coseno o la tangente de un 
ángulo
- Resolución de triángulos rectángulos, conocidos dos de sus lados,
o un lado y un ángulo agudo
- Representación de funciones trigonométricas
- Utilización de la trigonometría para la resolución de problemas geométricos reales
GUIÓN TEMÁTICO
Competencia en comunicación lingüística:
- Conocer, adquirir, ampliar y aplicar el vocabulario del tema
- Ejercitar una lectura comprensiva de textos relacionados con el núcleo temático
Competencia Matemática:
- Identificar razones trigonométricas
- Utilizar los algoritmos para resolver triángulos
- Resolver problemas matemáticos que involucren el uso de las razones trigono-
métricas y los teoremas del seno y coseno
Competencia en tratamiento de la información y competencia digital:
- Realizar las actividades propuestas haciendo uso de la calculadora y el ordenador
COMPETENCIAS
BÁSICAS
Las actividades propuestas se pueden utilizar como repaso, al final de la unidad, 
o intercalando las sesiones en segunda lengua una vez explicado los conceptos 
en la lengua materna.
Atención a la diversidad
Ampliación: WRITING WORD PROBLEMS
Refuerzo: USING YOUR CALCULATOR
OBSERVACIONES
Material didáctico en formato PDFFORMATO
Patricia Sánchez EspañaAUTORÍA
4 Material AICLE. 4º de ESO: Trigonometry
Tabla de programación AICLE
- Concebir el conocimiento científico como un saber integrado, que se estructura en 
distintas disciplinas, así como conocer y aplicar los métodos para identificar los prob-
lemas en los diversos campos del conocimiento y de la experiencia
- Comprender y expresarse en una o más lenguas extranjeras de manera apropiada
OBJETIVOS
- Razones trigonométricas de un ángulo
- Relaciones entre las razones trigonométricas
- Teoremas de Seno y del Coseno
- Resolución de triángulos rectángulos
- Funciones trigonométricas
TEMA
- Actividades para adquirir el vocabulario específico
- Ejercicios de cálculo de razones trigonométricas
- Relación de problemas de resolución de triángulos
- Presentaciones para el resto de compañeros en formato digital o en papel
TAREAS
- Reconocer y determinar las razones trigonométricas de un ángulo.
- Obtener razones trigonométricas con la calculadora.
- Utilizar la relación fundamental de la trigonometría.
- Hallar todas las razones trigonométricas de un ángulo a partir de una de ellas.
- Resolver un triángulo rectángulo, conociendo dos lados o un lado y un ángulo agudo.
- Aplicar la trigonometría en la resolución de problemas geométricos en la vida cotidiana.
CRITERIOS DE 
EVALUACIÓN
- Clasificar los tipos de razones trigonométricas
- Sintetizar y clasificar las diferentes fuentes de información
- Distinguir las partes de un triángulo rectángulo
- Analizar los diferentes tipos de triángulos
MODELOS
DISCURSIVOS
CONTENIDOS 
LINGÜÍSTICOS
FUNCIONES:
- Señalar partes de 
cuerpos geométricos
- Expresar resultados
- Preguntar sobre 
fórmulas matemáticas que 
han de aplicarse
ESTRUCTURAS:
Did you find....? 
Look for information
 Complete this chart with 
...
What does this expression 
mean?
I think this is the cosine.
I don’t think so. I agree. 
What is the theorem we 
have to use to solve this 
problem?
How do you read this?
LÉXICO:
Angle, right angle, arc, 
chord, radians, degrees, 
right triangle, sine, cosine, 
tangent, cosecant, secant, 
cotangent, sine rule, 
cosine rule, trigonometry 
function, ...
1. Contenidos comunes referentes a la resolución de problemas y la utilización de 
herramientas tecnológicas. 
4. Geometría.
CONTENIDOS
DE
CURSO / CICLO
5Material AICLE. 4º de ESO: Trigonometry
TRIGONOMETRY
What does … mean?
is there a formula to find 
this value?
What kind of real life problems 
can we solve with ...?
in my opinion the appropriate 
concept here is ... because... 
	
  
	
  
	
  
	
  
Any of these triangles could be solved using the Law of Sines and 
Cosines as long as we are given at least certain angles and sides.
6 Material AICLE. 4º de ESO: Trigonometry
VOCABULARY PRACTICE
1. Word Search. Find ten words and expressions 
related to trigonometry. Work in pairs.
Where did you put …?
I put it in …
… goes in …
Can you help me with ...,
I can’t find it.
How do you read this?
No, … does not go in …!
What does this word mean?
Can … be ...?
I don’t think so.
I agree
	
  
	
  
sine-function chord trigonometric-table angle triangle
angle-measure circle arc triangulation trigonometric-series
7Material AICLE. 4º de ESO: Trigonometry
2. The history of trigonometry
	
  
	
  
a) Listen
b) Read
The History of Trigonometry
The first _____________________ was apparently compiled by Hipparchus, who is now 
known as “the father of trigonometry.”
Ancient Egyptian and Babylonian mathematicians lacked the 
concept of an ______________________, but they studied the 
ratios of the sides of similar triangles and discovered some of 
the properties of these ratios. The ancient Greeks transformed 
trigonometry into an ordered science.
Ancient Greek mathematicians such as Euclid and Archimedes studied the properties of 
the ________ of an angle and proved theorems that are equivalent to modern trigonometric 
formulas, although they presented them geometrically rather than algebraically. The 
modern ___________________ was first defined in the Surya Siddhanta, and its properties 
were further documented by the 5th century Indian mathematician and astronomer 
Aryabhata. These Greek and Indian works were translated and expanded by medieval 
Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six 
trigonometric functions, had tabulated their values, and were applying them to problems 
in spherical geometry. 
At about the same time, Chinese mathematicians developed 
trigonometry independently, although it was not a major field 
of study for them. Knowledge of trigonometric functions and 
methods reached Europe via Latin translations of the works 
of Persian and Arabic astronomers. 
Driven by the demands of navigation and the growing need for 
accurate maps of large areas, trigonometry grew to be a major 
branch of mathematics. Bartholomaeus Pitiscus was the first to 
use the word, publishing his Trigonometria in 1595. Gemma Frisius 
described for the first time the method of _________________ 
still used today in surveying. It was Leonhard Euler who fully 
incorporated complex numbers into trigonometry. The works of 
James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential 
in the development of___________________.
	
  
	
  
	
  
8 Material AICLE. 4º de ESO: Trigonometry
3. Right triangle. Label the chart with the words on the right. Work in pairs.
a) Fill in the gaps with words from the word search.
b) Prepare a short summary of the text to present to your class
4. Trigonometric formulas. Match each concept with its formula. Work in pairs.
	
  
 
 
adjacent side 
opposite side 
hypotenuse 
angle 
right angle 
 
 
 
 
 
 
 
 
 
	
   	
   	
  
	
   	
   	
  
SINE
COSINE
TANGENT
COSECANT
SECANT
COTANGENT
Opposite / Adjacent
Opposite / Hypotenuse
Hypotenuse / Opposite
Adjacent / Hypotenuse
Adjacent / Opposite
Hypotenuse / Adjacent
9Material AICLE. 4º de ESO: Trigonometry
5. Sine and Cosine Rules. Listen to your teacher and write the rule.
	
  
	
  
SINE RULE
C
c
B
b
A
a
sinsinsin
==
side a divided by ___________________________________________
______________________________________________________________________________
COSINE RULE Abccba cos2222 −+= 	
  
side a squared equals ______________________________________
_________________________________________________________
______________________
10 Material AICLE. 4º de ESO: Trigonometry
6. Read the following text about trigonometric identities and underline 
the formulas you need in order to work on the trigonometric problems.
	
  
TRIGONOMETRY PRACTICE
An identity is an equality that is true for any value of the variable. An equation is an 
equality that is true only for certain values of the variable.
Reciprocal identities 
sin θ = 
1
 
 csc θ
csc θ = 1
 sin θ
 
cos θ = 
1
 
 sec θ
sec θ = 
1
 
 cos θ
 
tan θ = 
1
 
 cot θ
cot θ = 1 
 tan θ
Tangent and cotangent identities 
tan θ = 
sin θ
 cos θ
cot θ = 
cos θ
 sin θ
Pythagorean identities 
a) sin²θ + cos²θ = 1
b) 1 + tan²θ = sec²θ
c) 1 + cot²θ = csc ²θ
These are called Pythagorean identities, because they are the trigonometric version 
of the Pythagorean Theorem.
Note: sin² θ -- “sine squared θ “ -- means (sin θ)²
11Material AICLE. 4º de ESO: Trigonometry
7. Complete the following exercises on trigonometric functions. 
Explain your reasoning.
	
  
	
  
A. For the figure on the left, the value of sin C is
c / b
a / b 
b / a 
a / c 
c / a
B. For the figure on the right, the value of sin A + cos A is 
(b + c)/a 
(a + c)/b
(a + b)/c 
(a - b)/b 
(a + b + c)/b
D. For the figure on the right, which of the following relationships is true 
sin A = a / c 
cos A = b / c 
tan A = a / b 
sec A = b / a 
cot A = c / a
C. For the figure on the left, the value of cos C is
b / a 
a / b
c / a 
c / b 
b / c 
	
  
	
  
	
  
12 Material AICLE. 4º de ESO: Trigonometry
	
  
E. For the figure on the left, the value of cos C + sin A is
b/a + a/b 
a/b + c/b 
2a/b
2b/a 
b/c + c/a
F. Which of the following relationships is true
sin A / cos A = tan A
sin A / cosec A = cot A 
cos A / sin A = sec A 
cosec A / sin A = cos A 
tan A / cot A = sin A
H. (sin A / tan A) + cos A =
2 sec A 
sec A 
2 cosec A 
1 + cos A 
2 cos A
I. cot A tan A =
sin A 
Ο cos A 
sin A cos A 
1
1/(sin A cos A) 
G. tan A / sin A =
cosec A 
sec A
sin A 
cos A 
1 / sin A
13Material AICLE. 4º de ESO: Trigonometry
	
  
J. From the figure, the value of cosec A + cot A is
(a + b)/c 
a/(b + c) 
b/(a + c) 
(b + c)/a
(a + c)/b
K. Which of the following relationships is true
sin A cot A = 1 
cos A sec A = 1
sin A + cosec A = 1 
sec A - cos A = 1 
sec A cot A = 1
L. From the figure, the value of sin2 A + cos2 A is:
a/b + c/b
1
b/a + c/b
(a/b + c/b)2
(b/a + c/b)2 
N. cosec A / sec A =
cot A
tan A 
sin A 
cos A 
sin A + cos A
M. From the figure, the value of cot C + cosec C is
(a + c)/b 
(c + b)/a 
(a + b)/c
a/c + c/b 
c/a + b/c
O. For the figure on the right, the value of cot A is 
tan C
sin A / cos A 
cos C / sin C 
a / c 
c / b
	
  
	
  
	
  
14 Material AICLE. 4º de ESO: Trigonometry
8. The Sine Rule and the Cosine Rule. Complete the text below 
with the examples your teacher will give you.
We can use the laws of cosine and sine to solve any type of triangle.
Take two ratios: cross, multiply and rearrange to put the required quantity as the subject 
of the equation. 
The Sine Rule is useful to solve a triangle when the only given information is one angle 
and two sides if the angle is between the two sides, or two angles and one side.
Example 1
Example 2
Example 3
	
  
The Sine Rule
	
  
	
  
	
  
	
  
15Material AICLE. 4º de ESO: Trigonometry
The Cosine Rule means that in any triangle, the square of the length of one side is 
equal to the sum of the squares of the lengths of the other two sides, subtracting two 
times the product of the lengths of these sides and the cosine of the included angle.
The cosine rule is useful when you are given a triangle with three known sides, or one 
angle and two sides, where the angle is not between the two given sides.
There are two problem types:
 
1. You are given 2 sides + an included angle and need to work out the remaining side.
2. You are given all the sides and need to work out the angle.
Example 4
The Cosine Rule
	
  
	
  
Example 5
	
  
16 Material AICLE. 4º de ESO: Trigonometry
9. Complete the following activities using the sine and cosine 
rules. Show all your work.
	
  
A. Find the lengths of the other two sides (to 3 decimal places) of the triangles with
(a) a = 2, A = 30°, B = 40°
b = _________
c = _________
(b) b = 5, B = 45°, C = 60°
a = _________
c = _________
(c) c = 3, A = 37°, B = 54°
a = _________
b = _________
B. Find all possible triangles (give the sides to 3 decimal places and the angles to 
1 decimal place) with
(a) a = 3, b = 5, A = 32°
B = _________
C = _________
c = _________
(b) b = 2, c = 4, C = 63°
B = _________
A = _________
a = _________
(c) c = 2, a = 1, B = 108°
b = _________
A = _________
C = _________
17Material AICLE. 4º de ESO: Trigonometry
C. Find the length of the third side, to 3 decimal places, and the other two 
angles, to 1 decimal place, in the following triangles
(a) a = 1, b = 2, C = 30°
c = _________
A = _________
B = _________
(b) a = 3, c = 4, B = 50°
b = _________
A = _________
C = _________
(c) b = 5, c = 10, A = 30°
a = _________
B = _________
C = _________
D. Find the angles (to 1 decimal place) in the following triangles
(a) a = 2, b = 3, c = 4
A = _________
B = _________
C = _________
(b) a = 1, b = 1, c = 1.5
A = _________
B = _________
C = _________
18 Material AICLE. 4º de ESO: Trigonometry
10. Trigonometric Graphs. Compare and contrast the following graphs. 
Discuss their properties with your partner using the vocabulary given.
	
  
No Movement
 * were unchanged
 * did not change
 * remained constant
 * remained stable
 * stabilized
Prepositions
 * between 30º and 90º
 * from π to 2π
Movement: Down
 * fell
 * declined
 * dropped
 * decreased
 * sank
 * went down
Movement: Up
 * rose
 * went up
 * increased
 * grew
Tops and Bottoms
 * reached a peak
 * peaked
 * reached their highest level
 * fell to a low
 * sank to a trough
 * reached a bottom
Adjectives
 * slightly
 * a little
 * a lot
 * sharply
 * suddenly
 * steeply
 * gradually
 * gently
 * steadily
	
  
19Material AICLE. 4º de ESO: Trigonometry
11. Solve the following problems. Explain your reasoning.
	
  
a) A man is walking along a straight road. He notices the top of a tower subtending an 
angle A = 60º with the ground at the point where he is standing. If the height of the tower 
is h = 35 m, how far is the man from the tower?
b) A little boy is flying a kite. The string of the kite makes an angle of 30º with the ground. 
If the height of the kite is h = 15 m, find the length of the string that the boy is using.
	
  
	
  
20 Material AICLE. 4º de ESO: Trigonometry
	
  
c) Two towers face each other separated by a distance d = 40 m. As seen from the top 
of the first tower, the angle of depression of the second tower’s base is 60º and that of 
the top is 30º. What is the height of the second tower?
d) A ship of height h = 15 m is sighted from a lighthouse. From the top of the lighthouse, 
the angle of depression to the top of the mast and the base of the ship equal 30º and 45º 
respectively. How far is the ship from the lighthouse?
e) You are stationed at a radar base and you observe an unidentified plane at an altitude 
h = 1000 m flying towards your radar base at an angle of elevation = 30º. After exactly 
one minute, your radar sweep reveals that the plane is now at an angle of elevation = 60º 
maintaining the same altitude. What is the speed in m/s of the plane? 
	
  
	
  
	
  
21Material AICLE. 4º de ESO: Trigonometry
WRITING WORD PROBLEMS
12. Write 3 different word problems whose solutionsare based on the 
following graphs. Choose one to present to your class. Solve them.
	
  
	
  
	
   	
  
1. Peter can see ________
______________________
______________________
______________________
______________________
______________________
______________________
Solution 1. Solution 2. Solution 3.
2. We can see a boat ___
_____________________
_____________________
_____________________
_____________________
_____________________
_____________________
3. Peter can see ______
____________________
____________________
____________________
____________________
____________________
____________________
22 Material AICLE. 4º de ESO: Trigonometry
13. Chose one problem from those presented by your classmates, 
solve it and check your solution. 	
  
Problem: 
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________
Solution: 	
  
23Material AICLE. 4º de ESO: Trigonometry
USING YOUR CALCULATOR
14. Design a calculator in the template below and prepare a short 
presentation to share your design with the class. Locate the keys 
that will help you to find all the trigonometric functions.
	
  
Can you find the key to 
multiply / add / divide …?
Look, … is here. 
It’s next to …
Do you have this key?
Where is ….?
…. goes here.
24 Material AICLE. 4º de ESO: Trigonometry
15. Read the following information and practice with 
your calculator by completing the chart. 	
  
	
  
sin, cos, tan
Functions used in trigonometry concerning angles are sine (sin), cosine 
(cos), and tangent (tan).
- If you want to find the sine of an angle of 30 degrees, you enter 
30 and then the sin button. The answer should be 0.5.
inv
You can also go backwards. If you know the sine, cosine or tangent, 
you can find the angle in degrees for that function. This is called the 
inverse (inv) operation. 
Those values are also called the arcsine, arccosine and arctangent.
- Enter 0.5, click inv and then click sin. You should get 30 (degrees).
sinA cosA tanA secA cosecA cotA
0º
30º
45º
60º
90º
25Material AICLE. 4º de ESO: Trigonometry
16. Use your calculator to find the following angles - the trigonometric 
functions are given. Give your answer to the nearest tenth.
tan A = 1.23
A = _____ °
tan B = 2.56
B = _____ °
sin C = 0.78
C = _____ °
sin D = 0.527
D = _____ °
cos E = 0.352
E = _____ °
cos F = 0.725
F = _____ °
tan G = 0.786
G = _____ °
tan H = 1.275
H = _____ °
sin I = 0.468
I = _____ °
sin J = 0.867
J = _____ °
26 Material AICLE. 4º de ESO: Trigonometry
17. Develop a story involving the missing angle or missing side of a right triangle.
To support your story, you will need to draw or use a photograph or a picture 
clipped from a magazine or downloaded from the Internet, illustrating the problem 
in your story. Include all formulas and all of the steps required to solve your story.
Final Draft must include:
I. A creative, unique, and imaginative story 
• Imaginative 
• Appropriate subject
• Must be a trigonometry problem - finding a missing side or angle
II. A Drawing, an actual picture or a magazine clipping to illustrate your story
• From the Internet, magazine, newspaper, photograph 
• The drawing or picture must be visible and clearly definable
III. A diagram of the right triangle to solve the problem
• Provides a clear representation of the problem
• Include realistic measurements
• Include units
• It must be a right triangle, and labeled with a right angle mark
 
IV. Calculations
• Show formulas you used
• Show all steps
• Include units in your answer
 
V. An appealing presentation
• Professional look
• Story should be typed or neatly printed
• Colorful
• Appropriate presentation to your class
• Title
• Size limit: 8½ x 11 inches (21 x 28 cm)
FINAL PROJECT
	
  
The secret of the right triangle 
	
  
	
  
27Material AICLE. 4º de ESO: Trigonometry
SELF ASSESSMENT
Pictures taken from:
http://bancoimagenes.isftic.mepsyd.es/
Graphs taken from:
http://upload.wikimedia.org/wikipedia/commons/1/13/Sine_Cosine_Graph.png
ALWAYS SOMETIMES NEVER
LISTENING
I can understand my teacher talking about 
trigonometric functions
READING
I can understand problems related to triangles 
using the trigonometric rules
SPEAKING
I can talk about finding the solutions of 
problems related to triangles and trigonometric 
functions
WRITING
I can write the steps and solve problems 
related to triangles and trigonometric functions
VOCABULARY
I can use words related to trigonometric 
functions and rules