Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math10th grade LEARNING OBJECT LEARNING UNIT Using inverse trigonometric functions Trigonometry, using functions to study angle measurements. S/K Language Socio cultural context of the LO Curricular axis Standard competencies Background Knowledge Basic Learning Rights English Review topic Vocabulary box SCO 1: Find angles using trigonometric functions. SKILL 1: Identify the definition of inverse trigonometric functions. SKILL 2: Recognize the characteristics required to find the inverse of a function. SKILL 3: Find inverse functions using different representations. SKILL 4: Use calculators to find angle measurements from a given value of a trigonometric function. SKILL 5: Sketch graphs of inverse trigonometric functions using computer programs (graphing software, simulators, etc.). SKILL 6: Establish relationships between trigonometric functions and their inverse by comparisons. SKILL 7: Justify relationships between trigonometric functions and their inverse functions by using geometric arguments. English Neighborhood, school, city. Spatial thinking and geometrical systems. Describe and model periodic phenomena in the real world using trigonometric ratios and functions. Use geometric arguments to solve and propose real life problems from other subjects in mathematical contexts. Function, domain, range, ratio, trigonometric function, coordinate plane. Recognize shifts in graphs by variations in their functions: y = f(x) + a, y = b f(x), y = f(x+c), y = f(dx). Understand the definition of trigonometric functions: sine(x) and cos(x). The variable x can be any real number and the values of sine(x) and cos(x) are estimated based on calculations using the unit circle. βWHβ QUESTIONS Steep: level of inclination of a plane or path Accurate: Precise or correct Matter of seconds: the time needed is a few seconds NAME: _________________________________________________ GRADE: ________________________________________________ Introduction How steep is an inclined plane? Have you ever asked this when cycling up a mountain? 1 Image 1 Bicycle on an inclined plane These questions can be answered using mathematics, specifically: trigonometric inverse functions and some measurable information. This document will pay special attention to some characteristics of trigonometric functions. A function is the transformation of a given set of values into another given set of values. The boxes represent the sets and the arrow is the transformation. What inverse functions do is to invert the transformation process. 1 Retrieved from: https://pixabay.com/static/uploads/photo/2012/04/18/22/04/bicycle-38028__180.png To determine how steep is an inclined plane, it is necessary to define inverse trigonometric functions as well as graphing these functions with the use of graphing software that allows determining basic trigonometric ratios and delivering geometric arguments. Objectives ο· To relate inverse functions to find characteristics of trigonometric functions. ο· To find the measurement of an angle using inverse trigonometric functions. Activity 1 SKILL 1: Identify the definition of inverse trigonometric functions. SKILL 2: Recognize the characteristics required to fin the inverse of a function. SKILL 6: Establish relationships between trigonometric functions and their inverse by comparisons. Angle of inclination. What is an inverse trigonometric function? If in a right triangle we can find the length of a segment knowing the measurement of one acute angle and one of the other two sides, applying the inverse process means that one of the acute angles can be found knowing the relation between two sides. The inverse process is done using inverse trigonometric functions. When comparing the domain and range of a function and its inverse we can observe: Image shows how the domain of the inverse function π(π₯) = ππππ πππ (π₯) is the same as the range of π(π₯) = π πππ (π₯)and the domain of π(π₯) = π πππ (π₯) is the same as the range of the inverse functionπ(π₯) = ππππ πππ (π₯). The range of the inverse function is the domain of the original function and the domain of the inverse function is the range of the original function. For example, in the function π(π₯) = π πππ(π₯) with an angle of π/2 radians, the result is π π π ( 2 ) = π ππ ( 2 ) = 1 and the inverse function π(π₯) = ππππ πππ(π₯) with and angle of 1 radian, π the result isπ(1) = ππππ ππππ (1) = 2 . It Works the same with the other 5 trigonometric functions and their inverse. For example, π(π₯) = π‘ππππππ‘ (π₯), with a value of π 4 π 4 π 4 radians will result in π ( ) = π‘ππππππ‘ ( ) = 1, and the inverse function π(π₯) = ππππ‘ππππππ‘(π₯) with a value of 1 radian will result in π(1) = π ππππ‘ππππππ‘ (1) = 4 Consider this: In groups of three answer: Is it necessary to limit the domain to determine the inverse function? How is the domain of a function determined? An inverse function can exist only when it is a bijective function: a bijective function has one element in the domain that corresponds only to one element in the range, and for each element in the range there can only be one corresponding element in the domain. This is called a one-to-one correspondence and it is shown in the following image: Trigonometric functions are periodic and allow elements in the domain to have several π 3π images in the range. For example: πΆππ πππ 2 , has the same value as πΆππ πππ 2 restricting π π the domain to the interval [β 2 , 2 ] transforming it into a bijective function. Learning activity 1. Find the following words trigonometric, bijective. in the Word Search: inverse, domain, range, True β False questions 2. To have an inverse function, a function must have two characteristics: ο· (1) Every element in the domain must have an image in the range. ο· (2) Every element in the range must have a corresponding element in the domain. Label True (T) of False (F) the following statements for the function π¦ = π‘ππππππ‘ (π₯): a. Characteristic (2) does not belong to the function because for the values 5π 4 π 4 y in the domain, the image is the same in the range. (____________) b. Characteristic (1) π does not belong to the function because π ( 2 ) = ππππ πππ‘ ππ₯ππ π‘. (____________) c. Both characteristics are found in the function. (____________) d. Only characteristic (2) is found in the function. (____________) Activity 2 SKILL 3: Find inverse functions using different representations. SKILL 4: Use calculators to find angle measurements from a given value of a trigonometric function. SKILL 7: Justify relationships between trigonometric functions and their inverse functions by using geometric arguments. Representation of trigonometric functions and their inverse The representation of a function is the different ways certain information can be represented; an inverse function can be represented algebraically and graphically, on a coordinate plane, and more. The algebraic representations are: Trigonometric function y = sine (x) y = cosine (x) y = tangent (x) Representations β1 π¦ = π πππ (π₯) π¦ = πΆππ πππ β1 (π₯) π¦ = πππππππ‘ β1 (π₯) π¦ = ππππ πππ (π₯) π¦ = ππππππ πππ (π₯) π¦ = ππππ‘ππππππ‘ (π₯) The graphic representations shown below allow the comparison between the behavior of the function and its inverse. As seen before, the domain has been restricted so that there is an inverse function. COSINE Image 2 Graphic representation of cosine and its inverse For further detail, the following table shows the range of values of the domain and the range in image 4 f(x) = Cosine (x) Domain Range [0,Ο] [-1,1] f(x) = Arccosine (x) Domain Range [-1,1] [0,Ο] SINE Image 3 Graphic representation of sine and its inverse For further detail, the next table shows the range of values of the domain and the range in image 5. f(x) = Sine (x) Domain Range [-Ο/2,Ο/2] [-1,1] f(x) = Arccosine (x) Domain Range [-1,1] [-Ο/2,Ο/2] TANGENT Image 4 Graphic representation of tangent and its inverse For further detail, the next table shows the range of values of the domain and the range in image 6. f(x) = Tangent (x) Domain Range [-Ο/2,Ο/2] Real numbers f(x) = Arctangent (x) Domain Range [-β,β] [-Ο/2,Ο/2] Class activity: In groups of three, graph the trigonometric functions of secant, cosecant and cotangent and their inverse functions. Determine the domain and range of each of them and draw conclusions from the comparison. Share these conclusions with your classmates. Application of inverse trigonometric functions The characteristics and conditions required for there to be an inverse function have been established. Now you will see how inverse functions are applied in a real situation using an inclined planed; the slope of a mountain path. The inclined plane creates a right triangle which allows the use of trigonometric ratios (trigonometric ratios can only be used in right triangles). Remember that: trigonometric ratios are commonly known as the quotient of the sides of a right triangle. ππππ π = πππππ ππ‘π π πππ π»π¦πππ‘πππ’π π πΆππ πππ π = π΄πππππππ‘ π πππ π»π¦πππ‘πππ’π π πππππππ‘ π = πππππ ππ‘π π πππ π΄πππππππ‘ π πππ If the biker knows the length of the path is 25 meters and that the altitude will go from 1500 meters to 1515 meters above sea level, the following information can be found: ππππ π = 15 πππππ ππ‘π π πππ , 25 βπ¦πππ‘πππ’π π ππππ π = 0,6 π = ππππ πππ (0,6) Use your calculator to find the value of the angle which corresponds to the angle of inclination. 2 The mathematical process gives a result of 36.86° of inclination for the biker´s path. To find the complementary angle (sum of angles equals 90°) βΞ²β we use the following ratio: πΆππ πππ π½ = π΄πππππππ‘ π πππ π»π¦πππ‘πππ’π π The opposite side for angle π½ is 15 meters (height). Substituting the values: πΆππ πππ π½ = 15 25 πΆππ πππ π½ = 0,6 π½ = π΄πππππ πππ 0,6 2 Retrieved from: http://thumbs.dreamstime.com/x/calculadora-cient%C3%ADfica-12945025.jpg Created by the author Learning activity 1. Drag the trigonometric ratio to its equivalent. Trigonometric ratio ππππ π πΆππ πππ π πππππππ‘π π Equivalence ππππππππ‘ πππ‘βππ‘π’π βπ¦πππ‘πππ’π π πππππ ππ‘π πΆππ‘βππ‘π’π π΄πππππππ‘ πππ‘βππ‘π’π πππππ ππ‘π πΆππ‘βππ‘π’π π»π¦πππ‘πππ’π π 2. True or False a. The value of an angle can be found using an inverse trigonometric function. b. The inverse function of π(π₯) = πΆππ πππ (π₯), is known to be π¦ = ππππππ πππ β1 (π₯) 3. In pairs, write a text using your own words to explain why it is necessary to determine the interval of the domain in trigonometric functions so that there are the corresponding inverse functions? Activity 3 SKILL 5: Sketch graphs of inverse trigonometric functions using computer programs (graphing software, simulators, etc.). Inverse Trigonometric Functions on Graphing Software Geogebra and Desmos are online graphing software to graph any accurate inverse trigonometric functions in a matter of seconds. The steps π(π₯) = ππππππ πππ (π₯)to graph will be shown next: 1. Go on https://www.geogebra.org/algebra Image 5 Geogebra online platform 2. Write down the inverse function π(π₯) = ππππππ πππ (π₯) in the slot labeled βEntradaβ. Image 6 Writing down functions using Geogebra 3. Press βEnterβ and the software will graph the submitted function. Image 7 Inverse function graphs using Geogebra Using βDesmosβ (graphing software) the process is very similar. Use the inverse function π(π₯) = ππππππ πππ (π₯), 1. Go on https://www.desmos.com/calculator Image 8 Logging into the Desmos platform 2. Click on the plus symbol (agregar elemento) and select the first option βf(x) expressionβ. Image 9 Creating an element using Desmos 3. Use the functions keyboard to enter π(π₯) = ππππππ πππ (π₯) and graph it. The keyboard is a helpful tool to graph functions. Image 10 Desmos Functions Keyboard 4. After entering variable βxβ, the software will graph the function. Image 11 Inverse function graphed using Desmos Learning activity in class 1. Use the online graphing software to graph the following functions: π¦ = ππππππ πππ (π₯), π¦ = ππππ‘ππππππ‘ (π₯) y π¦ = ππππππ πππ (π₯) and determine: a. Domain. b. Range. c. How do the inverse function and the original function relate? Summary What is an inverse trigonometric function? Any function that allows finding the values of the domain (angles) using the values in the range. What characteristics do inverse functions have? It has to be a bijective function, meaning that it has two characteristics: How to find and angle using trigonometric functions? Using the trigonometric functionsπ(π₯) = π πππ (π₯), π(π₯) = πππ πππ (π₯) and π(π₯) = π‘ππππππ‘ (π₯), it is possible to find the angles using the ratios found in a right triangle. Replace the values of the segments and solve using inverse functions. ππππ π = πππππ ππ‘π π πππ π»π¦πππ‘πππ’π π πΆππ πππ π = π΄πππππππ‘ π πππ π»π¦πππ‘πππ’π π πππππππ‘ π = πππππ ππ‘π π πππ π΄πππππππ‘ π πππ For example: given the trigonometric expression ππππ π = 0,6 we solve for the angle π = ππππ π πππ (0,6) using a scientific calculator: π = 36, 86° What relationship is there between the domain and range of trigonometric functions and their inverse functions? The following are tables containing the domain and range of functions. f(x) = Sine (x) Domain Range [-Ο/2,Ο/2] [-1,1] f(x) = Arccosine (x) Domain Range [-1,1] [-Ο/2,Ο/2] f(x) = Tangent (x) Domain Range [-Ο/2,Ο/2] Real numbers f(x) = Arccotangent (x) Domain Range [-β,β] [-Ο/2,Ο/2] f(x) = Cosine (x) Domain Range [0,Ο] [-1,1] f(x) = Arco cosine (x) Domain Range [-1,1] [0,Ο] What tools (software) can be used to graph inverse trigonometric functions? Use graphing software to obtain accurate graphs in a matter of seconds. Two online graphing tools used to aid this process are: Geogebra and Desmos. Click on the following links: Geogebra https://www.geogebra.org/algebra Desmos: https://www.desmos.com/calculator Homework 1. Juan Carlos was flying kites and suddenly he wanted to know the angle of elevation of his kite. The information he gathered was that he had used 35 meters of rope and the horizontal distance between the kite and him was 180cm. Let´s help him answer the following questions: a) How can he find the angle of elevation? b) Based on your response to the previous question, why is it the best method to find the answer? 2. Determine the domain of π(π₯) = π πππππ‘π (π₯) that allows it to have an inverse function. Remember to show it is a bijective function. 3. Use the graphing software introduced in this learning guide to graph the following functions: π(π₯) = ππππππ πππππ‘ (π₯), π(π₯) = ππππ πππππ‘ (π₯), π(π₯) = πππππππ‘ππππππ‘ (π₯). Find the domain and range of each function. Evaluation Answer questions 1 and 2 labeling them as True (T) or False (F). 1. An inverse trigonometric function allows finding values from the range (angles) of a given function using values from the domain. 2. Trigonometric ratios are commonly known as the product of the sides in a right triangle. Choose the correct answer. 3. Which of the following options allows an inverse function for π(π₯) = π πππ (π₯)? I. Limit the domain because it is a periodic function that will have several elements in the domain and will share the same image. II. The function will have an inverse function if the domain is limited to the π π interval [ 4 , 2 ] III. IV. The function will have an inverse function if the domain is limited to the π π interval[β 2 , 2 ] . The range of the inverse function will be the interval [β1,1] because the values in the domain correspond to angle values. Choose the correct option: a. I and II b. I and III c. II and IV d. III and IV Match the correct options 4. Match the function to its inverse. Base function y = sine (x) y = cosine (x) y = tangent (x) Inverse function π¦ = πΆππ πππ β1 (π₯) π¦ = πππππ‘ππππππ‘ (π₯) π¦ = πππππ πππ (π₯) Fill in the blanks 5. Geogebra and Desmos are online ___________ software that graphs any accurate inverse trigonometric functions in a matter of ___________. 6. Match the basic trigonometric function to its inverse. Basic Function y = sine (x) y = cosine (x) y = tangent (x) Inverse π¦ = πΆππ πππ β1 (π₯) π¦ = ππππ‘ππππππ‘ (π₯) π¦ = ππππ πππ (π₯) Glossary Domain: the domain of a function or relation is the set of all possible independent values that a relation can have. It is the collection of all possible inputs. (Instituto de Monterey, 2010). Range: the range of a function or relation is the set of all possible values dependent relation can produce. It is the collection of all possible outputs.(Instituto de Monterey, 2010). Injective: It means that each element of the range "B" has at most one corresponding element of the domain "A" (but this does not tell us that all elements of "B" have any of "A"). (Disfruta Las Matemáticas.com, 2011). Surjective: It means that each element of the range "B" has at least one element of the domain "A" (maybe more than one). (Disfruta Las Matemáticas.com, 2011). Bijective: It means being injective and surjective at the same time. So there is a perfect match "one to one" relation between the elements of both sets. (Disfruta Las Matemáticas.com, 2011). Vocabulary Box Steep: level of inclination of a plane or path Accurate: Precise or correct Matter of seconds: the time needed is a few seconds Bibliography ο· Disfruta Las Matemáticas.com. (2011). Disfruta Las Matemáticas. Retrieved from: http://www.disfrutalasmatematicas.com/conjuntos/inyectivo-sobreyectivobiyectivo.html ο· Instituto de Monterey. (2010). Dominio y Rango. Retrieved from: https://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U03_L2 _T2_text_final_es.html