CONSEJERÍA DE EDUCACIÓN
Dirección General de Participación e Innovación Educativa
Identificación del material AICLE TÍTULO
Trigonometry
NIVEL LINGÜÍSTICO SEGÚN MCER
A2.2
IDIOMA
Inglés
ÁREA / MATERIA
Matemáticas
NÚCLEO TEMÁTICO
Geometría
GUIÓN TEMÁTICO
- 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
FORMATO
Material didáctico en formato PDF
CORRESPONDENCIA CURRICULAR
4º de Educación Secundaria. Matemáticas B
AUTORÍA
Patricia Sánchez España
TEMPORALIZACIÓN APROXIMADA
6 sesiones
COMPETENCIAS BÁSICAS
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 trigonométricas 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
OBSERVACIONES
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
Material AICLE. 4º de ESO:
Trigonometry
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Tabla de programación AICLE OBJETIVOS
- Concebir el conocimiento científico como un saber integrado, que se estructura en distintas disciplinas, así como conocer y aplicar los métodos para identificar los problemas en los diversos campos del conocimiento y de la experiencia - Comprender y expresarse en una o más lenguas extranjeras de manera apropiada
CONTENIDOS DE CURSO / CICLO
1. Contenidos comunes referentes a la resolución de problemas y la utilización de herramientas tecnológicas. 4. Geometría.
TEMA
- Razones trigonométricas de un ángulo - Relaciones entre las razones trigonométricas - Teoremas de Seno y del Coseno - Resolución de triángulos rectángulos - Funciones trigonométricas
MODELOS DISCURSIVOS
- 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
TAREAS
- 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
CONTENIDOS LINGÜÍSTICOS
CRITERIOS DE EVALUACIÓN
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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, ...
- 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.
Material 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.
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Trigonometry
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VOCABULARY PRACTICE Can you help me with ..., I can’t find it.
Can … be ...? I don’t think so. I agree
How do you read this?
No, … does not go in …! What does this word mean?
1. Word Search. Find ten words and expressions related to trigonometry. Work in pairs.
Where did you put …? I put it in … … goes in …
sine-function chord trigonometric-table angle triangle angle-measure circle arc triangulation trigonometric-series
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Material 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___________________.
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Trigonometry
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a) Fill in the gaps with words from the word search. b) Prepare a short summary of the text to present to your class
3. Right triangle. Label the chart with the words on the right. Work in pairs.
adjacent side opposite side hypotenuse angle right angle
4. Trigonometric formulas. Match each concept with its formula. Work in pairs.
SINE COSINE TANGENT COSECANT SECANT COTANGENT
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Material AICLE. 4º de ESO:
Trigonometry
Opposite / Adjacent Opposite / Hypotenuse Hypotenuse / Opposite Adjacent / Hypotenuse Adjacent / Opposite Hypotenuse / Adjacent
5. Sine and Cosine Rules. Listen to your teacher and write the rule.
a b c = = sin A sin B sin C
SINE RULE
side a divided by ___________________________________________ _________________________________________________________ _____________________
COSINE RULE
a 2 = b 2 + c 2 − 2bc cos A
side a squared equals ______________________________________ _________________________________________________________ ______________________
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Trigonometry
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TRIGONOMETRY PRACTICE 6. Read the following text about trigonometric identities and underline the formulas you need in order to work on the trigonometric problems.
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 = sec θ =
1 sec θ 1 cos θ
tan θ =
1 cot θ
cot θ =
1 tan θ
Tangent and cotangent identities sin θ tan 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
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Material 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
C. For the figure on the left, the value of cos C is
b/a a/b c/a c/b b/c
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
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Trigonometry
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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 G. tan A / sin A = cosec A sec A sin A cos A 1 / 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)
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Material 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 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
N. cosec A / sec A = cot A tan A sin A cos A sin A + cos A 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
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Trigonometry
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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.
The Sine Rule
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
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Material AICLE. 4º de ESO:
Trigonometry
The Cosine Rule
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
Example 5
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Trigonometry
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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 = _________
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Material 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 = _________
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Trigonometry
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10. Trigonometric Graphs. Compare and contrast the following graphs. Discuss their properties with your partner using the vocabulary given.
Prepositions * between 30º and 90º * from π to 2π
Movement: Down * fell * declined * dropped * decreased * sank * went down
Adjectives * slightly * a little * a lot * sharply * suddenly * steeply * gradually * gently * steadily
Material AICLE. 4º de ESO:
No Movement * were unchanged * did not change * remained constant * remained stable * stabilized
Tops and Bottoms * reached a peak * peaked * reached their highest level * fell to a low * sank to a trough * reached a bottom
Movement: Up * rose * went up * increased * grew
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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.
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Trigonometry
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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?
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Material AICLE. 4º de ESO:
Trigonometry
WRITING WORD PROBLEMS 12. Write 3 different word problems whose solutions are based on the following graphs. Choose one to present to your class. Solve them.
1. Peter can see ________ ______________________ ______________________ ______________________ ______________________ ______________________ ______________________
2. We can see a boat ___ _____________________ _____________________ _____________________ _____________________ _____________________ _____________________
3. Peter can see ______ ____________________ ____________________ ____________________ ____________________ ____________________ ____________________
Solution 1.
Solution 2.
Solution 3.
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Trigonometry
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13. Chose one problem from those presented by your classmates, solve it and check your solution.
Problem: _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________
Solution:
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Material AICLE. 4º de ESO:
Trigonometry
USING YOUR CALCULATOR Look, … is here.
…. goes here.
Do you have this key?
It’s next to …
Can you find the key to multiply / add / divide …?
Where is ….?
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.
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Trigonometry
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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
0º 30º 45º 60º 90º
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Material AICLE. 4º de ESO:
Trigonometry
secA
cosecA
cotA
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 = _____ °
cos F = 0.725 F = _____ °
tan B = 2.56 B = _____ °
tan G = 0.786 G = _____ °
sin C = 0.78 C = _____ °
tan H = 1.275 H = _____ °
sin D = 0.527 D = _____ °
sin I = 0.468 I = _____ °
cos E = 0.352 E = _____ °
sin J = 0.867 J = _____ °
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Trigonometry
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FINAL PROJECT 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
• 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)
Material AICLE. 4º de ESO:
The secret of the right triangle
V. An appealing presentation
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Trigonometry
SELF ASSESSMENT
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
Pictures taken from: http://bancoimagenes.isftic.mepsyd.es/ Graphs taken from: http://upload.wikimedia.org/wikipedia/commons/1/13/Sine_Cosine_Graph.png
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Trigonometry
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