Download Secuencia "With a little help from trigonometry"

WITH A LITTLE HELP OF
OBJECTIVES
At the end of this lesson, you will be able to:
1. Apply trigonometry to solve real world problems,
such as measuring the height of tall buildings
2. Identify the trigonometric ratio that must be used
3. Use the goniometer quadrant to measure angles
4. Apply Trig ratios to solve right-angled triangle problems
5. Use the calculator correctly
6. Define and describe processes correctly, in written
and oral form.
7. Represent each situation geometrically and
accurately
CONTENTS
1. Trigonometric ratios.
2. Use of the calculator to find the trigonometric ratios
(sinA, cos A and tanA).
3. Use of the calculator to calculate angles.
4. Use of formulae and trigonometric ratios to solve
right-angled triangle problems.
5. Use of trigonometry to calculate heights.
6. Use of formuale and trigonometric ratios to solve
everyday situations.
7. Use of instruments to measure angles
side
Sections:
1. Identifying and Defining
2. Calculations and Measures. Use
of Calculators.
3. Applications of Trigonometry
4. Writing and Solving Word
Problems (Card Game)
5. Measuring Heights: Our High
school
6. FINAL TASK: Using Trig to
measure city landmarks
ASSESSMENT CRITERIA
1 Use trigonometric ratios of angles less than 90º
correctly.
2. Use instruments of measure to calculate heights
3.Represent different situations by means of
geometric figures
4. Use the calculator correctly in order to solve
trigonometric ratios and calculate angles, knowing
the trigonometric ratio
5. Solve problems using the appropriate ratio
6. Describe the process followed to solve different
situations, accurately and using the specific
terminology
MATEMÁTICAS
1
SECTION 1: IDENTIFYING AND DEFINING
1.1 How can you measure or calculate the height of a really tall building? Read the text and
find out how these calculations were made in ancient times
The legend says that Egyptian priests wanted to test Thales of Miletus’ wisdom and asked him to solve
a really difficult task: to calculate the height of Cheops’ pyramid. There are others who say that it was the
Pharao Asmasis who called this geometrician to solve this difficult problem . Apparently, he wished to build
a pyramid taller than the pyramid of Cheops -- then, there was the need to know the measurements of this
pyramid. However, it was not an easy task: to measure the height of an oblique solid.
No matter how or why, the fact is that Thales (690 BC) used the following method to measure the Pyramid
height: he lay on the ground and marked the length of his own height with two sticks. Then, he stood up next
to one of these sticks and when he saw that his shadow measured the same as the length marked on the
ground, he measured the shadow of the famous pyramid. He told the priests: “Now that my shadow and my
height are the same, the length of the pyramid must be the same as its height.” So, he concluded its height
was 146.5 m. (488 ft).
As the distance from the Sun to the Earth is very big, the sun rays can be considered parallel when they reach
the pyramid and the person.
At present, it is completely unnecessary to wait for the sunset. We can use trigonometry to calculate the
height of the pyramid of Cheops. Maybe we won’t ever need to calculate the height of the pyramid of Cheops,
but there may be other everyday life situations where we may find it useful to apply the so called trigonometric
ratios to calculate the height of very high monuments, objects or buildings. In fact, we will use trig to find out
how high our school is.
1.2
2
Read again and complete the following statements:
a)
Thales of Miletus measured the height of the pyramid by ____________________
b)
Trigonometry is really useful to ________________________________________
c)
In order to calculate heights, we will use_________________________________
With a little help of trigonometry
SOLVING EVERYDAY PROBLEMS
1.3 How can we calculate heights?
Imagine that you have a tall tree in your yard that needs to be cut down. You want to make sure that the tree won't hit
your house when it falls. How can you approximate the height of the tree?
1.4 Watch the following video and see what these two people do.
http://www.learner.org/courses/learningmath/video/measurement/wmp/m5b1.html
1.5
Think-Pair-Share: What steps have been given?
1
2
3
4
5
6
MATEMÁTICAS
3
SECTION 2: CALCULATIONS AND MEASURES
In this section you will learn:
1. how to make simple calculations
2. the basics of trigonometry
DEVELOPING SKILLS
2.1 Measure the angles and sides of the right-angled triangles. Be careful when you take
measurements. If you want to get accurate results, you must take measurements correctly.
2.2 Fill in the table below. In the case of angle values, write just a decimal number; in the
case of sides, write two decimal numbers.
2.3 Share the results with your partner. Are they the same? Are they different? Why or Why
not?
4
With a little help of trigonometry
WRITING AND TALKING ABOUT MATHS
What happens to these quotients? Think-Pair-Share
DEVELOPING SKILLS
2.4 Time to measure, calculate and reach conclusions
a)
Choose an angle between 10º and 30 º
b)
Draw three right-angled triangles with the chosen angle
c)
Write the angles, measure the sides of these triangles and fill in the following
table. Follow instructions given in section number 2.
MATEMÁTICAS
5
WRITING AND TALKING ABOUT MATHS
d)
Why do you think the three quotients of the triangles are the same?
e)
What kind of triangles are they?
f)
Which of the following theorems will you use to demonstrate that the results of the quotients above are
the same?
g)
Pythagoras’ Theorem
Thales’ Theorem
The Height Theorem
The Sine Theorem
The remainder Theorem
The Cosine Theorem
What does this Theorem say? Write down its formulation and make a graph to explain it more clearly.
This Theorem says that _____________________________________________________________
________________________________________________________________________________
In this picture, ____________________________________________________________________
If
____________________________________________________________________________,
then _____________________________________________________________________________
Useful vocabulary:
Straight lines
Secant
Segment
Ratio
Quotient
Area
be equal to
Hypotenuse
Remember: Symbols and words.
It is important to use words and symbols appropriately. Part of being able to write
Mathematics well is knowing when to use symbols and knowing when to use words.
6
With a little help of trigonometry
DEVELOPING SKILLS
2. 5 Theory: naming the sides of a right -angled triangle
a) Read the text below and name the sides of these triangles
In a right triangle there are two acute angles,
two legs and one hypotenuse.
Leg
If we look at angle B, the two legs of the right
triangle will be called adjacent leg (AB leg,
next to angle B), and opposite leg (the AC
leg, opposite to angle B).
MATEMÁTICAS
7
2. 6 Theory: Trigonometric ratios
The ratio formed by the opposite leg of an acute angle and the hypotenuse is called the sine of the angle.
opposite leg
AC
sine of the angle B= sin B = ------------------------- =-------------=---------------------=
hypotenuse
BC
The ratio formed by the adjacent leg of an acute angle and the hypotenuse is called the cosine of the
angle.
adjacent leg
AB
cosine of the angle B= cos B = ------------------------- =-------------=---------------------=
hypotenuse
BC
Note: In a right triangle the measurements of the legs are less than the hypotenuse, so both trigonometric ratios
are less than one, 0 < cos B < 1 and 0 < sin B < 1
The ratio formed by the opposite leg of an acute angle and the adjacent leg is called the tangent of the
angle.
opposite leg
AC
tangent of the angle B= tan B = ------------------------- =-------------=---------------------=
adjacent leg
BC
Measure the sides and angles of the right triangle in section 2.5 and calculate the sine, cosine and tangent of the
two acute angles B and C.
opposite leg
AC
sine of the angle B= sin B = ------------------------- =-------------=---------------------=
hypotenuse
BC
adjacent leg
AB
cosine of the angle B= cos B = ------------------------- =-------------=---------------------=
hypotenuse
BC
opposite leg
AC
tangent of the angle B= tan B = ------------------------- =-------------=---------------------=
adjacent leg
BC
opposite leg
sine of the angle C= sin C = ------------------------- =-------------=---------------------=
hypotenuse
adjacent leg
cosine of the angle C= cos B = ------------------------- =-------------=---------------------=
hypotenuse
opposite leg
tangent of the angle C= tan C = ------------------------- =-------------=---------------------=
adjacent leg
8
With a little help of trigonometry
APPLYING SKILLS
2.7 Using a Calculator
a) Read the text and follow the instructions
Given an angle measure, find the sine or cosine ratio
On a calculator, we use the keys labelled SIN and COS to display the values of
the sine and cosine of an angle. The sequence of keys that a calculator requires
for tangent will be the same as the sequence for sine and cosine.
For example, to find sin 50º and cos 50º, we use the following:
ENTER SIN 50
) ENTER
ENTER COS 50 ) ENTER
DISPLAY
SIN (50)
0.766044
COS (50)
0.642787
Note: To make these calculations, make sure that the letter D (degrees) appear on the top of the calculator screen;
if not, press or type SHIFT MODE 3 (Deg.) . Now, the letter D will appear on the screen and you will able to check
the results shown in the example
b) Given the angles B and C which appeared on the previous section, use the calculator to
find the value of sin B, cos B, tan B, sin C, cos C and tan C.
Important: Check that the results coincide with the values obtained measuring the two sides of the triangle.
Given a sine or a cosine ratio, find the angle measure
-1
A calculator will also find the measure of <A when sin A or cos A is given. To do this we use the keys labeled SIN
and COS-1. These are the second function of SIN and COS and are accessed by first pressing SHIFT. We can
think of the meaning of sin-1 as “the angle whose sine is”. Therefore, if SIN A= 0.2588, then sin-1 (0.2588) can be
read as “the angle whose sine is 0.2588.”
To find the measure of angle A ,<A, from the calculator, we use the following sequences of keys:
E NT E R: SHIFT SIN - 1 O.2 5 8 8 ) EN TE R
DI S P LAY
SIN - 1 ( 0 .2 5 8 8 )
1 4 .9 9 8 8 7 0
Note: If we press the key o,,, we get the
angle in degrees, minutes and seconds; so
in this case we will get 14º 59’ 56’’
(the calculator shows 14º 59’ 55.93’’)
C) Use the calculator to fill in the table below, if possible.
sin A = 0.5
cos A = 0.6
tan A = 1
A=
A=
A=
cos A = 0.345
tan A = 34.6
cos A = 1.54
A=
A=
A=
MATEMÁTICAS
9
WRITING AND TALKING ABOUT MATHS
2.8 Using the definition below, as a model, define the rest of trigonometric ratios, cosine and
tangent of the right angle
In a right triangle ABC, B = 90º, the sine of the angle
A is defined as the quotient between the opposite leg
of the angle and the hypotenuse
opposite leg
a
sine of the angle A=sin A= ----------------------=-------hypotenuse
b
In right triangles, angles and their opposite sides are labelled
with the same letter; the only difference is that angles are
named with uppercase and sides with lowercase
REMEMBER:
a)
When talking about legs and/or hypotenuses, the triangles always have a right angle. So, in the definitions of the
trigonometric ratios, we must say “given a right triangle…”
b)
When solving problems using trigonometric ratios, we must make sure that we are applying them to right triangles.
10
With a little help of trigonometry
SECTION 3: APPLICATIONS OF TRIGONOMETRY
3.1 Match the following fields to the corresponding definition.
Geometry
It is the branch of mathematics concerned with the properties and relationships between points, lines, surfaces, solids and higher dimensional analogues.
Physics
It is the branch of science concerned with the nature and properties of matter
and energy.
Topography
It is a detailed description or representation on a map of the physical features of an area.
3.2 Listen to a talk on applications of trigonometry and find out the relationship existing
between these fields
What can trigonometry do for you?
3.3 Listen again and complete the following mindmap. Add as many boxes as necessary.
Trigonometry: its applications
Now, we will use trigonometry to solve some of the questions mentioned before
3.4.
Listen to the following word problems. You will hear them three times. Use the worksheet provided to take down notes, draw a sketch and solve them.
MATEMÁTICAS
11
Listening Comprehension Worksheet
Name:
Group:
Example nº ____
Data/details given
Sketch
Question(s):
o
_______________________________________
o
_______________________________________
Process and operations:
Solution/result:
12
With a little help of trigonometry
SECTION 4: WRITING AND SOLVING WORD PROBLEMS
Aims:
1. to work cooperatively in order to solve problems as quickly as possible
2. to be able to explain the process followed
3. to invent a word problem including the information provided
APPLYING SKILLS
A Card Game
Procedure:
1.
Form groups of 2 or 3 students
2.
Shuffle the cards.
3.
Distribute the cards among the groups but make sure that the teacher has got one of them.
4.
Each group must have a different card, with two different pieces of information: an answer to a word problem
and certain details which must be used to write their own math problem or question. First, you must write a word problem
including that information (data, phrases, words, etc…) The phrase in bold is the question to be calculated.
ANSWER
QUESTION
Roof of a building, angle of elevation = 45º, 5m closer the new angle of
elevation = 55º, Michael is 175cm, height of the building.
Height = 13.74cm
Unequal side= 14.62cm
5.
In 5 minutes, each group will have to INVENT and WRITE the word problem corresponding to the card given.
Here is an example
Michael, a photographer for the National Geographic, is trying to take a picture of a really beautiful building but he needs to know its exact height to adjust the tripod. From where he is standing, he can see the
roof of this building with an angle of elevation of 45º; if he moves 5m closer, the new angle is 55º. Knowing
that he is 175cm tall, calculate the height of the building
The problem must include not just the raw details. You must invent a situation, saying who, what, where
and/or why.
Here are some useful phrases and expressions:
Knowing that ______, calculate the ____________
Given _____, calculate the ________________
How far_______?
If ________________, how tall/high_______________?
6.
The first player is the teacher; he or she will read a text for the question which is on his or her card.
7.
You will try to solve the problem and see if you have the right answer on your card, by comparing the result you
obtained and the solution appearing in your card.
8.
If your group has the right solution in the card, you will be asked to solve the problem for the whole class. After
explaining to your partners how you made the calculations, you will read your own problem.
MATEMÁTICAS
13
QUESTION:
A barge is moving at a constant speed along a straight canal. The angle of elevation of a bridge is 10°.
After 10 minutes the angle of elevation is 15°.
After how much longer does the barge reach the bridge?
Give your answer to the nearest second.
EXPLANATION: AN EXAMPLE
If we call the speed in minutes “v”, then the distance travelled by the barge in 10
minutes is 10v metres, since d= speed*time=v *10= 10 v m
If we call “t” to the time it will take the barge to pass across the bridge, then the
distance travelled by the barge in t minutes is t v metres.
The graph or sketch of the situation is:
Since both the data and the unknown are on the legs of the right-angled triangle, we
must use the tangent ratio to solve this problem.
h
10v+ tv
h
tv
tan 10=-------------; tan15= ------------First, we find the value of “h” in both equations and match them
h= (10v+tv) tan 10= tv tan 15 a 10v tan 10+ tv tan10= tv tan 15
After that, we simplify v, and we obtain:
10 tan 10+ t tan 10= t tan 15
Finally, we find the value of the unknown t
10 tan10
10 tan 10= t (tan 15- 10) a t= -------------- = 19.245 min a t= 19 min 15 sec
tan15-tan10
Remember: you will have to explain the steps followed, so use linkers such as first, then, after that, finally.
9. Go on until all the cards have been read.
14
With a little help of trigonometry
SECTION 5: MEASURING HEIGHTS
Aim:
1. To follow instructions and make a goniometer quadrant
2. To apply trigonometry to the calculation of heights
5.1 The Process
a) But, first let’s watch the following video and learn about trigonometry and its functions. What is SOH-CAH-TOA? What
does it stand for? http://www.youtube.com/watch?v=jwGD5bY7dEA?
b) Watch again and Think-Pair-Share: What steps have been given to measure the height of the highest tree?
1
2
3
4
5
6
5. 2 Instruments to measure heights.
Throughout history, several instruments have been used to calculate the height of things. Here are just a few examples,
Can you recognize any of them? Match the names below to the corresponding picture.
What’s the missing name?
Jacob’s stick
a goniometer quadrant
a hypsometer
an astrolabe
a theodolite
MATEMÁTICAS
15
TASK: BUILDING AND USING GONIOMETER QUADRANTS
You will learn:
1. how to build a goniometer quadrant
2. how to use this instrument to measure the height of buildings (for example, the height of your
school)
Background information:
What is it? What is it like?
A goniometer is an instrument that allows us to measure angles. The term “goniometry” is derived from two Greek words,
“gonia”, meaning “angle” and “metron”, meaning
“measure”.
This quadrant is a tool used to measure angles in
vertical. It consists of a graduated arc and a plate
in the shape of a quarter of a circle. In one of the
sides there are two peepholes, to direct it towards
the right point.
Hanging from the vertex, there is a weight hanging,
which shows the vertical direction. The position of
the string of the weight on the graduated arc shows
the measurement.
Follow the instructions carefully and build a goniometer
How to construct a working goniometer model:
Things you will need:
•
A 20X20 cm square of thin wood
•
A cup and a nail or a push pin
•
Something heavy, which can be used as a weight (for
example, a nut or a screw)
•
A graduated semicircle
•
A hollow thin tube or two peepholes
Figure 1
1
2
16
Instructions:
First, make an enlarged photocopy of the graduated semicircle (you can also draw it on paper and cut out the
pattern). Remember: we only need a quadrant.
Secondly, paste or glue the cutout on the square of thin wood. See figure 1.
3
Firmly attach the nail right in the center of the graduated arc (Point A). Now, you can attach a weight to one end
of a string Attach the other end of the string to the push pin or nail
4
Finally, place/ attach the tube or the two peepholes on the top of the square (Points B and B´).
With a little help of trigonometry
Read the instructions carefully and find out how to use your goniometer.
Underline the key differences between the two procedures described.
How to use the goniometer quadrant
We will hold the quadrant in such a way that the
string hangs vertically, and we will place it so that
the highest part of what we want to measure can
be seen through the peepholes. Just then we will
look at the angle measure given by the string on
the graduated arc.
The procedure used to calculate heights with a goniometer quadrant varies depending on whether
the object to be measured has an accessible base
(for example, a tower) or not (a mountain, for instance).
When the object has an accessible base
Procedure:
When the object has an inaccessible base
Procedure:
•
First, we measure the distance from the object
base to the place where we will measure the angle using
the goniometer quadrant.
•
Next, we will use the quadrant to measure the angle A, that is the one corresponding to the highest part of
the object.
•
After that, we’ll use the trigonometric ratio (tan A),
so as to get the length value (y).
•
Finally, to obtain the height of the object, we must
add the height of the person who is using the quadrant to
the calculated figure (y +ho).
•
First, we will measure the distance from the object
base to the place where we will be standing to measure
the angle; besides, we will measure the angle (A) using the
goniometer quadrant the angle.
Be careful: you must wait till you see the top of the object to be measured.
•
Next, we will walk away from the object a specific
distance whose length (d) is known. Then, we will use again
the quadrant to measure the angle B, observing the highest
part of the object.
•
After that, we will use the trigonometric tangent
ratio with angles A and B to obtain a system of equations.
Once solved, it will give us the length value (y).
•
Finally, we must add the value y and the height
value of the person who is using the goniometer quadrant
(y +ho).
y
tan A= ---------d
y= d. tan A
h= y+ ho
Important: when we are calculating
measures, bear in mind that the three
points on the ground must be on a straight
line.
MATEMÁTICAS
17
Which one is correct? Why?
Extra: Click on the link below and practice measuring the height of important landmarks such as the Eiffel Tower
http://highered.mcgraw-hill.com/sites/0098840016/student_view0/chapter1/math_in_action_interactive.html
Task description: Calculate the height of our school, and prepare a report to
show your findings.
a) Driving question(s):
How tall is our school?
How can we measure its height?
What can Trig do for you?
b) Process:
In small groups,
1.
Use the goniometer quadrant to measure the height of our school. Make the calculations both for an accesible
and an inaccesible base.
Important: Pay attention to the measurement of the angles, as the more accurate these measurements are, the more accurate the height calculations will be.
2.
Try to think of other ways to calculate its height, but this time without using trigonometry. Try to describe three
other ways and apply them to measure the height of the school.
3.
What advantages do you think trigonometry has to calculate the height of objects, compared to the other methods?
18
4.
Check if the results you obtain using the different methods are approximate.
5.
Finally, prepare a report to show your calculations. Don’t forget to include tables and figures. (For dossier)
With a little help of trigonometry
ASSESSMENT CRITERIA
Criteria
Points
4
3
2
1
Explanation
A complete
response with a
detailed
explanation
Good solid
response with
clear explanation
Explanation is
unclear.
Misses key
points.
Use of Visuals
Clear diagram or
sketch with some
detail
Clear diagram or
sketch
Inappropriate or
unclear diagram
No diagram or
sketch.
Mechanics
No math errors
No major math
errors or serious
flaws in reasoning.
Demonstrated
knowledge
Shows complete
understanding
of the questions,
and processes.
Requirements
Goes beyond the
requirements of
the problem.
May be some seMajor math
rious math errors errors or serious
or flaws in
flaws in
reasoning.
reasoning.
Shows substantial Response shows Response shows
understanding of some understan- a complete lack
the problem, ideas,
ding of the
of understanding
and processes
problem.
for the problem
Meets the
requirements of
the problem
Hardly meets the
requirements of
the problem.
Does not meet
the requirements
of the problem.
TOTAL:
MATEMÁTICAS
19
FINAL TASK: Using Trig to measure city landmarks
Driving question(s):
How high is that famous landmark building?
What calculations and measurements will you have to do?
Process:
In groups (3 to 4 people), you will prepare a video or a presentation showing the procedure followed, the calculations
made and results obtained, when calculating the height of a building of your city.
1.
First, choose a landmark building or monument (tower, church, block of flats, etc.)
2.
Second, use the goniometer to calculate the height of this building.
Remember:
a)
You will have to take photos during all the process and present data, calculations and conclusions clearly.
b)
Every member of the group must take part both in the making and the presentation in class or video.
c)
You will have to hand in both the video or the presentation and the script used.
Duration of presentation: 6-8 minute
During the presentation, you will have to fill in the following form for each of the
groups
Team members
20
Landmark chosen
Explanation (1-10) Use of visuals (1-10)
With a little help of trigonometry
A FEW TIPS FOR YOUR PRESENTATION
Step 1 Planning and Preparation
Prepare! Prepare! Prepare!,
because preparation is everything.
Step 2 Structure
A good presentation has a clear structure, like a good book or film:
Introduction
Body
Conclusion
Step 3 Visual aids
(photos, images, tables, etc)
“A picture is worth 1,000 words”
Golden rule: use one image to give one message
Step 4 Signposting
When you read a book, you know where you are (the chapter, the page, the section,..)
but when you give a presentation, your audience does not know where they are- UNLESS
you tell them
Here are just a few useful expressions
My presentation is about …
First, I’d like to talk about ..
Second, …
This picture/slide/photo
shows
That’s the end of my presentation. Thank you for
listening. Have you got any
questions
Step 5 Delivery
Talk to the audience, don’t read and
make sure your audience can hear you clearly.
Remember:
use a loud and clear voice
speak naturally
make eye contact
vary pitch and tone
use appropriate words and grammar
Function
Introducing the subject
Starting one topic
Giving an example
Summarising and concluding
Ordering
Language
A few
Let’s begin by
words more use
fu
and p
First, …
hrase l
s
Now, we’ll move on to
Next,…
Let’s look now at..
For example
A good example of this is
In conclusion,
Let’s summarise
First, … second
Firstly, then, next, after that, finally
MATEMÁTICAS
21
ASSESSMENT CRITERIA
Criteria
Properly
organized to
complete project
Managed time
wisely
Acquired needed
knowledge
Process
Below Avg.
Satisfactory
Excellent
(1,2,3, 4)
(5, 6, 7)
(8,9,10)
(5, 6, 7)
(8,9,10)
(1,2,3, 4)
(1,2,3, 4)
(5, 6, 7)
Criteria
(8,9,10)
Product
Below Avg.
Satisfactory
Excellent
(1,2,3, 4)
(5, 6, 7)
(8,9,10)
Requirements of
the project.
(1,2,3, 4)
(5, 6, 7)
Organization and
structure
(1,2,3, 4)
(5, 6, 7)
(8,9,10)
(1,2,3, 4)
(5, 6, 7)
(8,9,10)
(1,2,3, 4)
(5, 6, 7)
(1,2,3, 4)
.
(5, 6, 7)
Explanation. Mechanics of writing
and speaking.
Maths mechanics
Use of visuals.
Creativity.
Cooperative work.
Participation.
Demonstrate
knowledge
(8,9,10)
(8,9,10)
(8,9,10)
(1,2,3, 4)
(5,6, 7)
With a little help of trigonometry
Points
(8,9,10)
TOTAL:
22
Points
.