Teaching & Learning Plans Plan 8: Introduction to Trigonometry Junior Certificate Syllabus

Teaching & Learning Plans Plan 8: Introduction to Trigonometry Junior Certificate Syllabus
Teaching & Learning Plans
Plan 8: Introduction to
Trigonometry
Junior Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have and also
to knowledge which may be necessary in order to support them in accessing this new
topic.
Learning Outcomes outline what a student will be able to do, know and understand
having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/or
Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching and
learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing the
topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on teacher input
and gives details of the key student tasks and teacher questions which move the
lesson forward.
ii.
Student Activities – Possible and Expected Responses: Gives details of
possible student reactions and responses and possible misconceptions students
may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Checking Understanding: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
Teaching & Learning Plan 8:
Introduction to Trigonometry
Aims
• To introduce the concept of trigonometry
• To understand the concept of sin, cos and tan
• To apply trigonometry to solve problems
Prior Knowledge
Students should have knowledge of the concept of ratio and should know that
division is not commutative. Students should be able to use a calculator to convert
fractions, correct to three decimal places, measure the lengths of sides of a triangle
to the nearest millimetre and draw to scale. Students should have studied Pythagoras’
Theorem and know the meaning of the term “hypotenuse”. Students may have studied
similar triangles in geometry – the fact that corresponding sides are proportional makes
trigonometry possible. Students will have understood how to calculate mean. Students
should know how to use a measuring tape and protractor.
Learning Outcomes
As a result of studying this topic, students will be able to
• correctly identify the hypotenuse in a right angled triangle, and the
opposite and adjacent sides of a given angle
• find a pattern linking the ratio of sides of a triangle with the angles and
hence understand the concepts of sine, cosine and tangent ratios of angles
• apply trigonometry to solve problems including those involving angles of
elevation and depression
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Teaching & Learning Plan 8: Introduction to Trigonometry
Relationship to Junior Certificate Syllabus
Sub-topics
Ordinary Level
Higher Level
(also includes OL)
Trigonometry
Solution of right-angled
triangle problems of a
simple nature involving
heights and distances,
including the use of the
theorem of Pythagoras.
Functions of 30˚, 45˚ and
60˚ in surd form derived
from suitable triangles.
Cosine, sine and tangent
of angles (integer values)
between 0˚ and 90˚.
Decimal and DMS values
of angles.
Solution of right-angled
triangles.
Resources Required
Calculators, measuring tapes, graph paper, cm rulers and a sunny day!
Introducing the Topic
Trigonometry is simply geometrical constructions where the ratio of the side lengths in
triangles is used to determine angular measurement.
Trigon, meaning triangle and metria, meaning measurement
Mathematicians have used trigonometry for centuries to accurately determine distances
without having to physically measure them (Clinometer Activity Appendix A). It can
also be used to calculate angles that would be very difficult to measure. Trigonometry
has uses in such areas as surveying, navigation, drawing and architecture.
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2
»» Students measure one
student’s height and the
length of his/her shadow
and fill in Student Activity
1A.
Outside Classroom
»» We are going to find the
angle of elevation of the
sun. Fill in Student Activity
1A.
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»» Students fill in Student
Activity 1D to find the
measure of some tall object
and check their answer
with other students in the
class.
»» Now using the angle of
elevation (from Student
Activity 1C) and the length
of the shadow of an object
whose height you cannot
physically measure (from
Student Activity 1B), again
use a scaled diagram to
determine the height of
the object.
»» From your scaled diagram
work out the angle of
elevation of the sun.
»» Students fill in Student
Activity 1C to find the
angle of elevation of the
sun and check their answer
with other students in the
class.
Inside Classroom
»» Fill in Student Activity 1C.
»» Students measure the
length of the shadow
of some tall object e.g.
flagpole or goalpost and
fill in Student Activity 1B.
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
KEY: » next step
• student answer/response
»» Can students identify the
angle of elevation?
»» Walk around observing
students as they take
measurements.
»» Circulate checking that
students can draw a scaled
diagram.
»» Have all students
understood the task so far?
Checking Understanding
»» Distribute Student Activity
1 (sunny day necessary!)
Teacher’s Support and
Actions
Lesson Interaction
Teaching & Learning Plan 8: Introduction to Trigonometry
3
»» Students can mark the
hypotenuse of each right
angled triangle.
»» Collaborate in pairs as you fill
in Student Activity 2.
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»» Mark the hypotenuse on the
triangles in Student Activity 2
of right-angled triangles.
• The longest side
• The side opposite the 90°
angle
»» What is the hypotenuse of a
right-angled triangle?
»» Why is the study of triangles
important?– show students
the PowerPoint images in
Appendix B.
Note: Later on, having learned
about the trigonometric ratios of sin,
cos and tan studentscan use these
trigonometric ratios to determine
the angle of elevation of the sun and
hence the height of for example a
tall tree or a goalpost.
KEY: » next step
»» Mark the hypotenuse on the board
with an arrow to show that the
side is opposite the right angle.
»» Draw a large right angled triangle
with the right angle marked.
»» Have the heading “Right angled
triangles” on the board. (RHS of
the board should have heading
“New Words”).
»» Distribute Student Activity 2.
• student answer/response
»» Are all students
able to identify
the hypotenuse?
»» Have all students
been able to
complete Student
Activity 1E?
»» Alternatively, you can use
ratios in similar triangles. Fill
in Student Activity 1E.
»» Students complete
Student Activity 1E.
Checking
Understanding
Student Learning Tasks: Teacher Student Activities: Possible Teacher’s Support and Actions
Input
and Expected Responses
Teaching & Learning Plan 8: Introduction to Trigonometry
4
• 90°
• Complementary angles
• The hypotenuse
»» How many degrees do they
add up to?
»» What are these angles called?
»» Mark either one of the two
complementary angles on the
triangle with an arc. What is
the name of one of the arms
of that angle?
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»» The other side, which is beside »» Students mark one angle
with an arc and label
the marked angle, is given the
name “adjacent”. Label it.
the adjacent side to this
angle on all the triangles.
• Two
»» How many other angles are in
the triangle?
KEY: » next step
»» Circulate checking that students
can label the sides correctly;
emphasising that the labelling
of a side depends on the angle
it refers to. Adjacent means
adjacent to a particular angle.
»» Point out on the board that
regardless of which of the two
angles adding up to 90° is chosen
by the student, the hypotenuse is
one of the arms of that angle.
»» Tell students that the word
adjacent means “beside”. They
are sitting beside or “adjacent to”
another student.
»» Label the adjacent and write
the word “Adjacent” in the new
words list on RHS of board.
»» Label the hypotenuse on the
triangle on the board.
Student Activities: Possible Teacher’s Support and Actions
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 8: Introduction to Trigonometry
• student answer/response
»» Can all students
identify the side
adjacent to the
marked angle?
Checking
Understanding
5
»» If a student cannot
explain the answer, give
two lengths and then ask
the student if the order
matters.
»» Divide the length of one
side by the length of
another side using the
same units.
»» Yes, because 3/2 is not the
same as 2/3 for instance.
»» If you were to work out
the ratio of any two of
the sides in a right angled
triangle how would you do
this?
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»» Are students familiar
with the concept of ratio
and do they understand
that division is not
commutative?
»» Ask students to think
about the answer first and
then ask a student for an
answer.
»» Students label 90° angle,
the hypotenuse, one other
angle then adjacent and
then opposite sides.
»» Repeat this process of
marking angles and
labelling sides for all
the triangles on Student
Activity 1.
»» Would the order matter?
Explain your answer.
»» Are all sides correctly
labelled in each triangle?
»» Circulate checking the
work of each group.
• If you were standing at the
vertex of the marked angle
it would be across from
you or opposite you so it is
called the opposite side.
»» Describe the opposite side.
KEY: » next step
• student answer/response
»» Check each group’s work as »» Are all sides correctly
they are labelling the sides
labelled in each triangle?
of the triangle.
»» Students label the side
opposite the marked angle
as the opposite.
Checking Understanding
»» Label the third side of the
triangle as the opposite
side.
Teacher’s Support and
Actions
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 8: Introduction to Trigonometry
6
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»» Students work on
Student Activities 3, 4, 5,
6, 7 and 8.
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»» One student is to measure, one
to calculate ratios and one to
double check changing roles on
each triangle.
»» Student Activities 3, 4, 5, 6,
7 and 8: For each triangle
of the five triangles mark
the 90° angle and one other
given angle (given on the
sheet). Label the sides of the
right angled triangles as hyp
(hypotenuse), adj (adjacent),
and opp (opposite). Measure
each side correct to the nearest
mm and calculate the ratios
opp/hyp, adj/hyp, opp/adj.
• One is the inverse of
the other. If they were
multiplied, they would
»» If I knew the answer to the first
give an answer of 1.
ratio was ½, what would be
the answer to the second one? • 2/1
»» What is the relationship
between the first pair?
»» When giving the ratios use the
names for the sides.
»» How many possible ratios could • 6
be worked out for a rightangled triangle?
• student answer/response
»» Are students measuring
accurately and
calculating ratios
correctly i.e. opp/hyp
and not hyp/opp?
KEY: » next step
»» Emphasise that students
should measure accurately to
the nearest mm.
»» Distribute Student Activity 3,
4, 5, 6 ,7 and 8 to different
groups of 3-4 students each
i.e. one Student Activity per
group.
»» Circle
and
tell students that you will
concentrate on these 3 as the
others are their multiplicative
inverses.
»» Do students know
the concept of a
multiplicative inverse?
»» Write the ratios on the board »» Are the students able
as students call them out
to take the 3 sides and
placing multiplicative inverses
combine them into 6
beside each other.
pairs?
Student Learning Tasks: Teacher Student Activities: Possible Teacher’s Support and Actions Checking Understanding
Input
and Expected Responses
Teaching & Learning Plan 8: Introduction to Trigonometry
7
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•
»» If so, which one or which
ones and why?
- The opposite is
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bigger than the adjacent
when it is opposite
the bigger of the 2
complimentary angles.
• Yes
• Numerator will always
be smaller than the
denominator as the
hypotenuse is the longest
side.
•
»» Is it possible for any of the
ratios to be bigger than 1?
»» Explain.
»» Of the 3 ratios which of
them can never be bigger
than 1?
?
• student answer/response
»» Do students understand
that the only one of the
three ratios which can be
bigger than 1 is
»» Have they been able to
answer this question based
on their knowledge of
right angled triangles?
»» Are the students finding
that the ratios are
unchanged for a particular
angle regardless of the size
of the triangle?
Checking Understanding
KEY: » next step
»» If students cannot answer
this question, give an
example using numbers
when fractions give
answers greater than 1 or
less than 1.
»» Circulate checking the
progress of the different
groups to see that
sides are being labelled
correctly and that students
understand the task.
»» Ask the class when they
have a few of the ratios
calculated to answer this
question and then ask one
group to explain.
• The ratios are unchanged
»» If a group has difficulty
for a particular angle
seeing the pattern or
regardless of the size of the
verbalising it you will see
triangle.
this as you circulate and
help by using leading
questions.
»» Write down what you
have observed from your
answers.
Teacher’s Support and
Actions
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 8: Introduction to Trigonometry
8
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»» Each group gives the
answers for each ratio
for the angle they have
worked on.
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»» On the master table,
Student Activity 9 fill in
the mean value you have
calculated for sin, cos and
tan of the angle from your
own Student Activity 3, 4,
5, 6, 7, or 8. Tell the rest
of the class the values you
have calculated.
• student answer/response
»» Have all students
understood the tasks
so far, completed them
for all the triangles, and
filled out the master
table?
Checking Understanding
KEY: » next step
»» Draw a master table on the
board and fill in the answers
as they are given telling
students that they will be able
to check those answers in the
next step.
»» Hand out Student Activity 9.
»» Go back to Student
»» Beside each ratio students
»» Tell students that sin is the
Activities 3 - 8 and fill in
fill in the appropriate name
short version of sine, cos is
the name of each ratio, for
plus the angle it refers to.
short for cosine and that tan
example for the sheet with
is the shortened version of the
angles of 30° fill in opp/hyp
word ‘tangent’.
= sin 30° etc.
»» Students write this heading »» Write the word
and the ratios into their
TRIGONOMETRY on the board.
copies.
»» Tell the students that the
ratios they have investigated
have special names and write
them on the board.
»»
»» We have been studying
how the sides and angles
of triangles are related
to each other. This is
called TRIGONOMETRY –
Trigon meaning triangle
and metria meaning
measurement.
Teacher’s Support and Actions
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 8: Introduction to Trigonometry
9
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»» Circulate to ensure that
students are able to see the
patterns and are using correct
terminology.
»» Distribute Student Activity
10.
»» Circulate to see that the
values of sin, cos and tan of
angles calculated through
measurements agree with
those found on the calculator.
»» Emphasise that sin, cos and
tan are functions of angles.
»» Show students how to check
if their calculator is in degree
mode and, if not, how to put
it into this mode.
Teacher’s Support and Actions
KEY: » next step
• Students may be familiar with »» Write on the board - Given a
trig ratio, for example sin A
the SHIFT or 2nd function
= 0, the angle A = sin-1(0.5).
button on the calculator
and hence suggest using this
Emphasise that sin-1x, cos-1 x
button.
and tan-1 x represent angles
where x is a ratio of sides in a
right angled triangle.
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»» If you knew the ratios how
would you find out the
angle? Given that the sin of
an angle is 0.5 how do you
find the angle?
»» Using the answers on Student »» Students see patterns in the
answers on the master table.
Activity 9, answer the
questions on Student Activity
10.
»» Using the calculator check
the values of sin, cos and
tan of the angles, which you
have calculated through
measurement. Check the
measurements of the rest of
the class also.
»» Students check the values
in the master table with
the values got using the
calculator and fill in to
“check” columns on the
master table Student Activity
9.
• Degrees
»» What units are angles
measured in?
»» There are other units for
measuring angles such as
radians, which you will
meet later on so you must
be sure your calculator is in
degree mode if you are using
degrees.
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 8: Introduction to Trigonometry
• student answer/response
Checking
Understanding
10
• Trigonometry is about the study of
the relationship between angles and
ratio of sides in triangles.
Reflection
»» List what you have learned
today.
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»» Write down any questions
you may have.
»» Write down anything you
found difficult.
»» Students use their calculators to
evaluate these angles.
»» Given
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• The three ratios of sides in a right
angled triangle are:
• The sides in a right angled triangle
are labelled hypotenuse, and then
adjacent and opposite depending
on which of the two complementary
angles is of interest.
Student Activities: Possible and
Expected Responses
Student Learning Tasks:
Teacher Input
KEY: » next step
»» Circulate and take
note particularly of
any questions students
have and help them to
answer them.
Teacher’s Support and
Actions
Teaching & Learning Plan 8: Introduction to Trigonometry
• student answer/response
»» Are students using
the terminology with
understanding?
Checking Understanding
11
Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 1
Me and my shadow
Safety warning: Never look directly at the sun
Name: _________________________________________ Class: _____________________________
Date: _________________________________________ Time: ______________________________
When the sun is high, your shadow is short.
When the sun is low, your shadow is long.
Student Activity 1A
• Show the angle of elevation of the sun on the above diagram. Call it A.
• Describe the angle of elevation of the sun in terms of the two arms of the
angle._________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
• Measure the height of one of the students in your group and the length of their shadow.
• Height of the student: __________cm. Length of the shadow _____________________cm.
• Draw a rough sketch of a right-angled triangle to model the situation and write in the measurements.
Student Activity 1B
Measure the length of the shadow of some tall object e.g. flagpole or goalpost. Length
of the shadow of a tall object which you cannot physically measure e.g. goalpost
______________________________cm
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 1C
Back in class – Measuring the angle of elevation of the sun
• Decide what scale to use.
• Draw an accurate diagram on graph paper.
Diagram 1
• Measure the angle of elevation of the sun from Diagram 1 above using a protractor.
• Angle of elevation of the sun at _________ (time) on _______ (date) was ________.
• Check your answer with other students in the class.
• If you were to measure the angle of elevation of the sun at 10 a. m and another class measured the
angle at 11 a.m. what would be the difference in the measurements?_______________________________
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 1D
Knowing the angle of elevation of the sun, measure
the height of a tall object using the length of its
shadow as previously measured.
• Decide what scale to use. Scale: ______________________
• Draw an accurate diagram on graph paper using the length of the shadow, the
angle of elevation of the sun and forming right-angled triangle (ASA).
Diagram 2
• Measure the height of the goalpost from Diagram 2 above and using the scale factor convert to its
actual height.
• Check the answer with other students in the class.
Conclusion for part 2: The height of the goalpost is ___________________cm approximately.
Would you expect the same answer if you took the measurements at different times of the day?
Explain your answer.________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 1E
Using Similar Triangles
• Using graph paper draw the above 2 diagrams overlapping, with the angles of
elevation of the sun superimposed as shown by example in the diagram on the right.
Label the diagram on the graph paper as in the diagram on the right.
• What do you notice about the 2 vertical lines in the triangles? ______________________________________
_______________________________________________________________________________________________
• Measure the heights of the 2 vertical lines |ED| and |CB|.
• Measure the 2 horizontal lines |AB| and |AD|.
____
_____
• What do you notice about the two ratios? ________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
• Knowing |AB| and |CB| and the distance |AD| how could you find |ED| without knowing the angle of
elevation |∠EAD| of the sun?
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 1E
Ratios in Similar Triangles
• Draw 3 different right angled triangles with the arms of the 90 degree angle being vertical and
horizontal line segments, using the same angle of elevation which you calculated for the sun.
Call the triangles T1, T2, T3.
Measure the length of the vertical and horizontal line segments in these triangles.
Vertical T1
Horizontal T1
Vertical T2
Horizontal T2
Vertical T3
Horizontal T3
What do you notice about the ratios of any 2 vertical line segments of any 2 of these triangles and the
ratio of the corresponding horizontal line segments?_ ______________________________________________
_________________________________________________________________________________________________
_________________________________________________________________________________________________
_________________________________________________________________________________________________
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 2
Labelling Sides in Right Angled Triangles
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 3
Calculating ratios for similar right angled triangles with angles of 30°
• Measure and label the 90° and the 30° angles in the following triangles. What is the measure of the
third angle?
• Label the hypotenuse as “hyp”. With respect to the 30° angle, label the other sides as “adj” for
adjacent and “opp” for
• Complete the table below.
Marked
Angle
Size=30°
|opp|/mm |hyp|/mm
|adj|/mm
(for angle=30°)
(for angle=30°)
fraction decimal fraction decimal fraction decimal
(for angle=30°)
T1
T2
T3
T4
T5
Mean Value (correct to 2 decimal places)
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 4
Calculating ratios for similar right angled triangles with angles of 40°
• Measure and label the 90° and the 40° angles in the following triangles. What is the measure of the
third angle?
• Label the hypotenuse as “hyp”. With respect to the 40° angle, label the other sides as “adj” for
adjacent and “opp” for opposite.
• Complete the table below.
Marked
Angle
Size=40°
|opp|/mm |hyp|/mm
|adj|/mm
(for angle=40°)
(for angle=40°)
fraction decimal fraction decimal fraction decimal
(for angle=40°)
T1
T2
T3
T4
T5
Mean Value (correct to 2 decimal places)
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 5
Calculating ratios for similar right angled triangles with angles of 45°
• Measure and label the 90° and the 45° angles in the following triangles. What types of right angled
triangle are these triangles?___________________________________________________________________
• Label the hypotenuse as “hyp”. With respect to the 45° angle, label the other sides as “adj” for
adjacent and “opp” for opposite.
• Complete the table below.
Marked
Angle
Size=45°
|opp|/mm |hyp|/mm
|adj|/mm
(for angle=45°)
(for angle=45°)
fraction decimal fraction decimal fraction decimal
(for angle=45°)
T1
T2
T3
T4
T5
Mean Value (correct to 2 decimal places)
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 6
Calculating ratios for similar right angled triangles with angles of 50°
• Label the 90° and the 50° angles in the following triangles. What is the measure of the third angle?
______________________________________________________________________________________________
• Label the hypotenuse as “hyp”. With respect to the 50° angle, label the other sides as “adj” for
adjacent and “opp” for opposite.
• Complete the table below.
Marked
Angle
Size=50°
|opp|/mm |hyp|/mm
|adj|/mm
(for angle=50°)
(for angle=50°)
fraction decimal fraction decimal fraction decimal
(for angle=50°)
T1
T2
T3
T4
T5
Mean Value (correct to 2 decimal places)
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 7
Calculating ratios for similar right angled triangles with angles of 60°
• Measure and label the 90° and the 60° angles in the following triangles. What is the measure of the
third angle?__________________________________________________________________________________
• Label the hypotenuse as “hyp”. With respect to the 60° angle, label the other sides as “adj” for
adjacent and “opp” for opposite.
• Complete the table below.
Marked
Angle
Size=60°
|opp|/mm |hyp|/mm
|adj|/mm
(for angle=60°)
(for angle=60°)
fraction decimal fraction decimal fraction decimal
(for angle=60°)
T1
T2
T3
T4
T5
Mean Value (correct to 2 decimal places)
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 8
Calculating ratios for similar right angled triangles with angles of 70°
• Measure and label the 90° and the 70° angles in the following triangles. What is the measure of the
third angle?___________________________________________________________________________________
• Label the hypotenuse as “hyp”. With respect to the 70° angle, label the other sides as “adj” for
adjacent and “opp” for opposite.
• Complete the table below.
Marked
Angle
Size=70°
|opp|/mm |hyp|/mm
|adj|/mm
(for angle=70°)
(for angle=70°)
fraction decimal fraction decimal fraction decimal
(for angle=70°)
T1
T2
T3
T4
T5
Mean Value (correct to 2 decimal places)
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 9
Master table of class results for ratios of sides in
right angled triangles
Angle/°
Check
Check
Check
30°
40°
45°
50°
60°
70°
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Teaching & Learning Plan 8: Introduction to Trigonometry
Student Activity 10
Using the master table of class results answer the following questions
1. What do you notice about sin 30° and cos 60°? _ __________________________________
________________________________________________________________________________
2.What do you notice about cos 30° and sin 60°?_ ___________________________________
________________________________________________________________________________
3.Can you explain what you have noticed using diagrams?
4.How would you describe angles 30° and 60°? _____________________________________
________________________________________________________________________________
5.Can you find similar examples in the master table? ________________________________
________________________________________________________________________________
6.For what angle in a right angled triangle is the opposite side one half of the
hypotenuse? _ __________________________________________________________________
________________________________________________________________________________
Draw a diagram to illustrate your answer.
7.For what angle in a right angled triangle are the opposite and adjacent sides equal? _
________________________________________________________________________________
8.Calculate
for each angle A. Compare this to the value of Tan A. What do you
notice? Can you justify the answer?_______________________________________________
________________________________________________________________________________
________________________________________________________________________________
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Teaching & Learning Plan 8: Introduction to Trigonometry
Appendix A
Making and using a clinometer to find the
height of a tall structure
Materials required: Protractor, sellotape, drinking straw, needle and thread, paper clip.
Drinking Straw
Thread
h1
A
h2
Finding the height of a wall/spire/ flagpole using a clinometer
Work in threes – one holding the clinometer, one
reading the angle of elevation, one recording the
angle of elevation.
• Measure the height of the observer from eye to
ground level.
• Measure the distance from the observer to the
base of the building (under the highest point).
• Mark the position of the observer on the
ground.
© Project Maths Development Team 2009
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• Hold the clinometer so that the string is vertical.
• Now tilt the clinometer looking through the
drinking straw so that the highest point on the
top of the wall/flagpole/spire is visible.
• Read the angle of elevation of this highest
point to the nearest degree.
• Draw a rough sketch of the situation marking
in the distances measured and the angle of
elevation.
26
Teaching & Learning Plan 8: Introduction to Trigonometry
Appendix A
Rough Sketch
Height from ground to
observer’s eye
Distance from observer to the
foot of the spire
Angle of elevation to the top
of the spire
Back in class:
• Calculate the height of a very tall object using a scaled diagram.
• Using graph paper draw a scaled diagram of the above situation. Scale_______________
• Measure the height of the spire from the scaled diagram and using the scale factor convert to its actual
height.
• Height of the spire above the observer’s eye: ____________________
• Height from ground to the observer’s eye: ______________________
• Height of the spire: ___________________________________________
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27
Teaching & Learning Plan 8: Introduction to Trigonometry
Appendix A
Calculate the height of a very tall object
using trigonometry
Redraw the diagram (doesn’t have to be to scale) marking the right angle, the hypotenuse, the
angle of elevation, the side adjacent to the angle of elevation and the side opposite the angle of
elevation. Right angled triangle, sides labelled and measurements included
• What side do you know the length of? _________________________________________
• What side do you require the length of? ________________________________________
• What trigonometric ratio in a right-angled triangle uses these 2 sides? ________________
Using trigonometry, calculate the height of the building.
Angle of
elevation of
the top of the
building
Distance to
the base of
the building d
Tan A = h1/d
h1
Height of the
observer h2
Height of the
building
h1+ h2
Question
As you move towards or away from the building while sighting the top of the spire, the angle
of elevation of the top of the spire varies. What angle of elevation would allow the height of
the building to be measured by using the distance from the observer to the base of the building
added to the observer’s height – i.e. no scaled drawing or trigonometry required?
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28
Teaching & Learning Plan 8: Introduction to Trigonometry
Appendix A
Finding the height of a building with a
moat around it
h1
A
B
h2
d
Work in threes – one holding the clinometer, one reading the angles of elevation, one recording
the angles of elevation.
• Measure the height of the observer from eye to ground level, h2, and fill into the table.
h2
⏐<A⏐
⏐<B⏐
⏐d⏐
h1
H = h1+h2
• Mark the position of the observer on the ground.
• Hold the clinometer so that the string is vertical.
• Now tilt the clinometers, looking through the drinking straw so that a point on the top of the wall/
flagpole/spire is visible.
• Read the angle of elevation, <A, to the nearest degree and fill into the table.
• The observer moves closer to the building and again views the top of the building through the
drinking straw on the clinometer.
• Read the angle of elevation, <B, to the nearest degree and fill into the table.
• Measure the distance between the two viewing positions, d, of the observer and fill into the table
below.
• Draw a rough sketch of the situation marking in the distances measured and the angles of elevations.
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29
Teaching & Learning Plan 8: Introduction to Trigonometry
Appendix A
Use a scaled diagram to calculate the height
of the building with a moat
Using graph paper draw a scaled diagram of the above situation. Scale _____________________
• Measure the height of the spire from the scaled diagram and using the scale factor convert to its actual
height.
• Height of the spire above the observer’s eye: _____________________
• Height from ground to the observer’s eye: _______________________
• Height of the spire: ______________________________________
© Project Maths Development Team 2009
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30
Teaching & Learning Plan 8: Introduction to Trigonometry
Appendix A
Calculating the height of a very tall object
surrounded by a moat using trigonometry
Redraw the diagram (which does not have to be to scale) labelling the sides and angles.
Right angled triangle, sides labelled and measurements included
•
•
•
•
•
•
•
Fill in all the angle and side measurements known for triangles CBD and CAB.
Fill in the other two angle measurements in triangle CAB.
Of the two triangles CBD and CAB, which triangle do you have most information for? ______________
Which side do you require the length of in triangle CBD? _________________________________________
What side is shared by both triangles CAB and CBD? ______________________________________________
What rule can be used to calculate this side? _____________________________________________________
Calculations:
•
•
•
•
•
Label the sides in right angled triangle CBD appropriately as ‘hypotenuse’, ‘adjacent’ and ‘opposite’.
Which side do you know the length of? _________________________________________________________
Which side do you require the length of ? _______________________________________________________
What trigonometric ratio in a right-angled triangle uses these 2 sides? _____________________________
Calculations to find the required length:_________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
• Complete the table below to find the height of the building H
⏐<A⏐
© Project Maths Development Team 2009
⏐<B⏐
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⏐d⏐
h2
h1
H = h1+h2
31
Teaching & Learning Plan 8: Introduction to Trigonometry
Appendix B
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32
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