Download Math10th grade LEARNING OBJECT Using inverse trigonometric

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.
“WH” 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

To relate inverse functions to find characteristics of trigonometric functions.

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 𝑓(𝑥) = 𝑎𝑟𝑐𝑠𝑖𝑛𝑒 (𝑥) is the same as the
range of 𝑓(𝑥) = 𝑠𝑖𝑛𝑒 (𝑥)and the domain of 𝑓(𝑥) = 𝑠𝑖𝑛𝑒 (𝑥) is the same as the range of the
inverse function𝑓(𝑥) = 𝑎𝑟𝑐𝑠𝑖𝑛𝑒 (𝑥). 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 𝑓(𝑥) = 𝑠𝑖𝑛𝑒(𝑥) with an angle of 𝜋/2 radians, the result is
𝜋
𝜋
𝑓 ( 2 ) = 𝑠𝑒𝑛 ( 2 ) = 1 and the inverse function 𝑓(𝑥) = 𝑎𝑟𝑐𝑠𝑖𝑛𝑒(𝑥) with and angle of 1 radian,
𝜋
the result is𝑓(1) = 𝑎𝑟𝑐𝑠𝑖𝑛𝑒𝑜 (1) = 2 .
It Works the same with the other 5 trigonometric functions and their inverse. For
example, 𝑓(𝑥) = 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (𝑥), with a value of
𝜋
4
𝜋
4
𝜋
4
radians will result in 𝑓 ( ) = 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 ( ) = 1,
and the inverse function 𝑓(𝑥) = 𝑎𝑟𝑐𝑡𝑎𝑛𝑔𝑒𝑛𝑡(𝑥) with a value of 1 radian will result in 𝑓(1) =
𝜋
𝑎𝑟𝑐𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (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: 𝐶𝑜𝑠𝑖𝑛𝑒 2 , has the same value as 𝐶𝑜𝑠𝑖𝑛𝑒 2 restricting
𝜋 𝜋
the domain to the interval [− 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 – False questions
2. To have an inverse function, a function must have two characteristics:

(1) Every element in the domain must have an image in the range.

(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 𝑦 = 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (𝑥):
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 𝑓 ( 2 ) =
𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡. (____________)
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
−1
𝑦 = 𝑠𝑖𝑛𝑒 (𝑥)
𝑦 = 𝐶𝑜𝑠𝑖𝑛𝑒 −1 (𝑥)
𝑦 = 𝑇𝑎𝑛𝑔𝑒𝑛𝑡 −1 (𝑥)
𝑦 = 𝑎𝑟𝑐𝑠𝑖𝑛𝑒 (𝑥)
𝑦 = 𝑎𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 (𝑥)
𝑦 = 𝑎𝑟𝑐𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (𝑥)
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,π]
[-1,1]
f(x) = Arccosine (x)
Domain
Range
[-1,1]
[0,π]
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
[-π/2,π/2]
[-1,1]
f(x) = Arccosine (x)
Domain
Range
[-1,1]
[-π/2,π/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
[-π/2,π/2]
Real numbers
f(x) = Arctangent (x)
Domain
Range
[-∞,∞]
[-π/2,π/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.
𝑆𝑖𝑛𝑒 𝜃 =
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐶𝑜𝑠𝑖𝑛𝑒 𝜃 =
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑇𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 =
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
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:
𝑆𝑖𝑛𝑒 𝜃 =
15 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
,
25 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑆𝑖𝑛𝑒 𝜃 = 0,6
𝜃 = 𝑎𝑟𝑐𝑠𝑖𝑛𝑒 (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°) “β” we use the following
ratio:
𝐶𝑜𝑠𝑖𝑛𝑒 𝛽 =
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
The opposite side for angle 𝛽 is 15 meters (height). Substituting the values:
𝐶𝑜𝑠𝑖𝑛𝑒 𝛽 =
15
25
𝐶𝑜𝑠𝑖𝑛𝑒 𝛽 = 0,6
𝛽 = 𝐴𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 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
𝑆𝑒𝑛𝑜 𝜃
𝐶𝑜𝑠𝑒𝑛𝑜 𝜃
𝑇𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝜃
Equivalence
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑐𝑎𝑡ℎ𝑒𝑡𝑢𝑠
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐶𝑎𝑡ℎ𝑒𝑡𝑢𝑠
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑐𝑎𝑡ℎ𝑒𝑡𝑢𝑠
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐶𝑎𝑡ℎ𝑒𝑡𝑢𝑠
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
2. True or False
a. The value of an angle can be found using an inverse trigonometric function.
b. The inverse function of 𝑓(𝑥) = 𝐶𝑜𝑠𝑖𝑛𝑒 (𝑥), is known to be 𝑦 = 𝑎𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 −1 (𝑥)
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 𝑓(𝑥) = 𝑎𝑟𝑐𝑐𝑜𝑠𝑒𝑛𝑜 (𝑥)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 𝑓(𝑥) = 𝑎𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 (𝑥) in the slot labeled “Entrada”.
Image 6 Writing down functions using Geogebra
3. Press “Enter” and the software will graph the submitted function.
Image 7 Inverse function graphs using Geogebra
Using “Desmos” (graphing software) the process is very similar.
Use the inverse function 𝑓(𝑥) = 𝑎𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 (𝑥),
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 “f(x)
expression”.
Image 9 Creating an element using Desmos
3. Use the functions keyboard to enter 𝑓(𝑥) = 𝑎𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 (𝑥) and graph it. The keyboard
is a helpful tool to graph functions.
Image 10 Desmos Functions Keyboard
4. After entering variable “x”, 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: 𝑦 = 𝑎𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 (𝑥),
𝑦 = 𝑎𝑟𝑐𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (𝑥) y 𝑦 = 𝑎𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 (𝑥) 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 functions𝑓(𝑥) = 𝑠𝑖𝑛𝑒 (𝑥), 𝑓(𝑥) = 𝑐𝑜𝑠𝑖𝑛𝑒 (𝑥) and 𝑓(𝑥) = 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (𝑥), 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.
𝑆𝑖𝑛𝑒 𝜃 =
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐶𝑜𝑠𝑖𝑛𝑒 𝜃 =
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑇𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 =
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
For example: given the trigonometric expression 𝑆𝑒𝑛𝑜 𝜃 = 0,6 we solve for the angle 𝜃 =
𝑎𝑟𝑐𝑜 𝑠𝑒𝑛𝑜 (0,6) using a scientific calculator:
𝜃 = 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
[-π/2,π/2]
[-1,1]
f(x) = Arccosine (x)
Domain
Range
[-1,1]
[-π/2,π/2]
f(x) = Tangent (x)
Domain
Range
[-π/2,π/2]
Real numbers
f(x) = Arccotangent (x)
Domain
Range
[-∞,∞]
[-π/2,π/2]
f(x) = Cosine (x)
Domain
Range
[0,π]
[-1,1]
f(x) = Arco cosine (x)
Domain
Range
[-1,1]
[0,π]
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 𝑓(𝑥) = 𝑠𝑒𝑐𝑎𝑛𝑡𝑒 (𝑥) 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: 𝑓(𝑥) = 𝑎𝑟𝑐𝑐𝑜𝑠𝑒𝑐𝑎𝑛𝑡 (𝑥), 𝑓(𝑥) = 𝑎𝑟𝑐𝑠𝑒𝑐𝑎𝑛𝑡 (𝑥), 𝑓(𝑥) = 𝑎𝑟𝑐𝑜𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (𝑥). 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 𝑓(𝑥) = 𝑠𝑖𝑛𝑒 (𝑥)?
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[− 2 , 2 ] .
The range of the inverse function will be the interval [−1,1] 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
𝑦 = 𝐶𝑜𝑠𝑖𝑛𝑒 −1 (𝑥)
𝑦 = 𝑎𝑟𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (𝑥)
𝑦 = 𝑎𝑟𝑐𝑜𝑠𝑖𝑛𝑒 (𝑥)
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
𝑦 = 𝐶𝑜𝑠𝑖𝑛𝑒 −1 (𝑥)
𝑦 = 𝑎𝑟𝑐𝑡𝑎𝑛𝑔𝑒𝑛𝑡 (𝑥)
𝑦 = 𝑎𝑟𝑐𝑠𝑖𝑛𝑒 (𝑥)
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

Disfruta Las Matemáticas.com. (2011). Disfruta Las Matemáticas. Retrieved from:
http://www.disfrutalasmatematicas.com/conjuntos/inyectivo-sobreyectivobiyectivo.html

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