Download TRIGONOMETRIC FUNCTIONS Teacher`s Guide

TRIGONOMETRIC
FUNCTIONS
Teacher's Guide
Trigonometric Functions 1: Solving Triangles
Teacher's Guide
The teacher's guide is designed for use with the television series Trigonometric
Functions 1: Solving Triangles. The programs are broadcast by TVOntario, the
television service of The Ontario Educational Communications Authority. For
broadcast dates consult the appropriate TVOntario schedule. The programs are
available on videotape. Ordering information for videotapes and this publication
appears on page 42.
The Series
Producer/Director:
Project Officers:
David Chamberlain
John Amadio
David Way
The Guide
Project Leader:
Author:
Editor:
Designer:
David Chamberlain
Ron Carr
Mei-lin Cheung
Roswita Busskamp
© Copyright 1989 by The Ontario Educational Communications Authority.
All rights reserved.
Printed in Canada
3414/89
CONTENTS
Introduction
3
Program One:
Trigonometric Ratios
Program Two:
Solving Right Triangles
11
Program Three:
Angles on the Plane
18
Program Four:
The Sine Law
24
Program Five:
The Cosine Law
31
Program Six:
Applications of Sine and Cosine Laws
36
5
INTRODUCTION
Trigonometry is a study of angle measurement. Rather than simple measurement as with the use of a protractor in plane geometry, it is calculation
using special functions which depend on angles and which are called
trigonometric functions.
The six-program Trigonometric Functions 1: Solving Triangles defines these
functions using the lengths of the sides of a right-angled triangle, and using
relationships on a coordinate plane. Using a calculator, or tables, the
measures of angles and the lengths of sides in triangles are calculated. Two
special relationships, the Sine Law and the Cosine Law, are developed, and
practical applications related to solving triangles are presented.
Trigonometric Functions 1: Solving Triangles, as in all the series in the Concepts in Mathematics group, uses the contemporary 3-D computer animation introduced in the first series, Vectors. The study of trigonometry is continued with special reference to the properties of trigonometric graphs in
the companion series, Trigonometric Functions 2.
PROGRAM ONE:
TRIGONOMETRIC RATIOS
Objectives
After viewing this program and completing several suggested activities
and problems, students should be able to do the following:
1. Locate the hypotenuse, the opposite side, and the adjacent side with
respect to a given acute angle in a right-angled triangle.
2. Use the Pythagorean relationship to calculate the length of one side of
a right-angled triangle, given the lengths of the other two sides.
3. Define the primary trigonometric ratios (sine, cosine, and tangent) for
an acute angle in a right-angled triangle.
4. Use tables and calculators to find the values of the trigonometric ratios
for a given angle.
5. Use an appropriate trigonometric ratio to find the measure of an angle
or the length of a side in a right-angled triangle.
b. "Solve" right-angled triangles.
7. Solve word problems involving right-angled triangles.
Previewing Activities
1. This program assumes that students are familiar with basic terminology related to angles and triangles. Spend some time reviewing concepts and definitions of the following: acute, obtuse, right-angled
triangle (right triangle), scalene, oblique, equilateral, angle-sum =180 °.
2. Review the use of the protractor to find the measures of angles in
degrees.
3. Review the Pythagorean relationship for the lengths of the sides of
right-angled triangles. A number of examples should be done using
tables of square roots and/or calculators.
4. In most cases, calculations result in answers which are correct to one
decimal place. Review this concept and discuss other methods of expressing results (significant digits, for example).
Program Description
This program opens with a view of a rocket-launch facility where Probe 1,
a spacecraft designed for cosmic exploration, is preparing for blast-off. We
are reintroduced to our astronaut friends, Ed and Charlie, who are in
control of the spacecraft, and looking forward to exploring the universe.
As they proceed, the measure of an angle must be found so that the
astronauts may continue to be in communication contact with Mission
Control.
In order to define the primary trigonometric ratios, a right-angled triangle
is introduced. The sides of this triangle are "named" with respect to a given
acute angle. The terms "hypotenuse," "opposite side," and "adjacent side"
are discussed. The trigonometric ratios, sine, cosine, and tangent of an acute
angle in a right-angled triangle are defined, and appropriate notation is
discussed.
The program shows how the measure of an acute angle is calculated when
the lengths of the three sides are known. Trigonometric tables and the use
of a calculator are introduced.
The new skills are applied to the transmission problem experienced by Ed
and Charlie. A triangle is created and, using trigonometry, the angle
needed to lock their signal with CapCom is calculated. Communication
contact is resumed and our space travellers continue on their journey.
Postviewing Activities
1. Since the sine, cosine, and tangent of an acute angle have been defined
as ratios of the lengths of the hypotenuse, opposite side, and adjacent
side, students should practise locating these sides in right-angled
triangles. Triangles should be used in varying orientations.
2.
The definitions of the primary trigonometric ratios must be memorized.
Values for the sine, cosine, and tangent of an acute angle can be
calculated using a number of different right-angled triangles and
knowing the lengths of all three sides.
3.
Values for the ratios should be found using right-angled triangles in
which the lengths of two sides are given. Using the Pythagorean relationship, the length of the third side is calculated first and then the
values of the ratios are found.
4.
Some time should be spent showing students that the values of the
trigonometric ratios depend on the measure of the acute angle, not on
the size of the triangle. This can be accomplished by using similar right
angled triangles and finding values of the ratios for corresponding
angles.
The use of calculators and tables should be practised. Values of trigonometric ratios such as sin 65°, cos 10°, and tan 47.5° can be found. As well,
tables or a calculator may be used to find an angle, given the value of the
ratio. For example, given sin
6.
Students should practise finding the length of a side of a right-angled
triangle, given the measure of an angle and the length of one side. A
suggested approach is to have the students form a ratio
"unknown" side
known side
and then determine the trigonometric ratio that this represents for the
given angle. For example:
The equation is then solved.
Find the lengths of the indicated sides in the following triangles.
b
10
27.5
7. The following examples involve division. For example:
5 = tan 38°
x
x
5
tan 38°
This may be an appropriate point, depending on the progress of the
class, to introduce the definitions of the reciprocal (secondary) trigonometric ratios.
cosecant 0
= 1 __
sin 0
hypotenuse
opposite side
secant 0
_ hypotenuse
= 1
cos 0 _ adjacent side
_ adjacent side
cotangent 0 = 1
tan 0 - opposite
side
Find the lengths of the indicated sides in the following triangles.
8.
A
To "solve" a triangle means to find the lengths of any "unknown" sides
and the measures of any "unknown" angles. Students should be
reminded that they can use skills from trigonometry, as well as the
Pythagorean relationship and the angle sum of a triangle, where
appropriate. Solve the following triangles.
9. Students should practise solving word problems involving right-angled
triangles. In each case, the student should make a rough sketch of the
triangle to illustrate the given and required information, and then draw
a reasonably accurate right-angled triangle representing the question.
(a) A telephone pole is supported by a number of guy-wires which are
attached to the top of the pole and to concrete footings on the
ground. One of the wires is 20.3 m long and makes an angle of 40°
with the level ground. Find the height of the pole.
(b) A 10-m ladder reaches 7.4 m up a wall. How far is the foot of the
ladder from the wall? Find the angle that the ladder makes with the
wall.
(c) From the observation deck of a lighthouse, the angle of depression of a rowboat in the bay is 7°. If the deck is 25 m above water
level, how far is the boat from the lighthouse?
PROGRAM TWO:
SOLVING RIGHT TRIANGLES
Objectives
After viewing this program and completing the suggested activities and
problems, students should be able to do the following:
1. Define and explain the meaning of an "angle in standard position" in a
Cartesian Plane, using terms such as "vertex," "initial arm," and
"terminal arm."
2. Find the length of a line segment in the plane, and thus find the distance
from the origin to a point on the terminal arm of an angle.
3. Define the primary trigonometric ratios for an angle in standard position with point (x,y) on the terminal arm and radius r units.
4. Find the values of the three primary ratios for angles in standard
position.
5. Become aware of angles larger than 90° and angles less than 0° (direction
of rotation), and use calculators to find the values of their trigonometric
ratios.
6. Work with problems in which two or more right-angled triangles are
involved.
Previewing Activities
1. Before students view this program, some time should be spent reviewing some items from the previous program. The naming of the hypotenuse, opposite and adjacent sides in a right-angled triangle should be
practised, as should the definitions of the sine, cosine, and tangent
ratios of an acute angle.
2. Review methods used to find the lengths of sides and the measures of
angles in right-angled triangles (Pythagoras, angle-sum =180° , trigonometric ratios).
3. Practise finding appropriate trigonometric ratios to apply to "solve"
triangles.
4.
In this program the Cartesian Coordinate Plane is introduced. Students
should review this system including the scales on the axes, ordered
pairs, and the naming of points using positive and negative real
numbers.
5.
If time permits, a teacher may wish to review a method of finding the
distance between two points on the Cartesian Plane by constructing a
right-angled triangle and applying the Pythagorean relationship.
Program Description
When we left Ed and Charlie in Program One, they has just solved their
problem of how to transmit and receive messages from their home base on
earth. Using trigonometry, they found that they must aim their transmissions at an angle of 37° to the vertical. This program opens with Mission
Control advising that, due to increased danger from meteors, the fixed
transmission satellite GEOSAT4 would be moved to a higher orbit. Ed and
Charlie must move farther away from earth as well, and they realize that
new calculations must be made.
Ed and Charlie's predicament is used as an example to review the application of trigonometric ratios to solve right-angled triangles.
The program then moves to a discussion regarding angles in the Cartesian
Coordinate Plane. A brief review of naming points using ordered pairs is
presented. The concept of an angle in standard position is introduced with
vertex at (0,0), initial arm along the positive X-axis, and the terminal arm
somewhere in the plane. An angle is defined in terms of a rotation from a
fixed position.
Using an example in which the point P(3,5) lies on the terminal arm, the
length of the line segment OP is found using the Pythagorean relationship.
The primary trigonometric ratios, sine, cosine, and tangent of an angle in
standard position, with point P(x,y) on the terminal arm and with OP = r
units, are defined.
Values of trigonometric ratios for angles with terminal arms in other
quadrants are calculated.
The program ends with Ed and Charlie suitably impressed by this new
information. They plan to continue their journey to the black hole, but in his
excitement, Ed presses the "Don't Ever Touch" button, sending Probe 1
spinning away, uncontrollably.
Postviewing Activities
1.
The definitions of the primary trigonometric ratios as related to an angle
in standard position must be memorized and practised. Students
should calculate these values for angles with terminal arms in varying
positions.
Y
Y
Y
2.
Calculations should be made for situations in which the value of either
x or y is not known.
3.
Using an appropriate trigonometric ratio and a calculator, find the
measure of angles a and Q in the following diagrams.
4.
If students have been introduced to the reciprocal trigonometric ratios,
it would now be appropriate to discuss the definitions of the cosecant,
secant, and cotangent ratios of an angle in standard position.
Y
Some time should also be spent discussing the importance of the restrictions. Since the value of r is always positive, the sine and cosine ratios
are defined for all real values. However, with the other four ratios, the
denominator may be zero, producing an undefined value for the ratio.
5. Students may be interested in why we seem to always rotate in a
counterclockwise direction. It should be explained that, from its initial
position on the positive X-axis, the terminal arm may rotate in either
direction. By convention, mathematicians have decided that, if the
rotation is counterclockwise, the angle will be considered positive. If
the direction of rotation is clockwise, the angle is considered to be
negative.
Draw diagrams illustrating the following angles in standard position:
45°, 255°, -60°, -180°, 420°.
6. Angles whose terminal arms lie in the same position are called coterminal angles. For example, 45° and -315° are co-terminal angles.
Find three angles which are co-terminal with 75 °. (Remember that you
can rotate clockwise and counterclockwise, and that you can rotate
more than one revolution.)
How many angles are co-terminal with 75°?
7. Some problems requiring the application of trigonometry involve more
than one right-angled triangle. In the following two-step situations,
find the lengths of the indicated sides.
PROGRAM THREE:
ANGLES ON THE PLANE
Objectives
After viewing this program and completing the suggested activities and
problems, students will be able to do the following:
1. Given an angle between 90° and 360°, express it in the form (180° ± 0)
or (360° - 0 ).
2. Find and express the relationships
sin (180° - 0) = sin 0
cos (180° - 0) = - cos 0
3. Find the sign of a trigonometric ratio using the CAST rule.
4. Use a calculator to find the measures of angles with terminal arms in
quadrants II, III, and IV.
5. Find the primary trigonometric ratios of an angle with its terminal arm
on a coordinate axis (0°, 90°,180°, 270°, 360°).
Previewing Activities
1. Before viewing this program, students should review the definitions of
the primary trigonometric ratios for an angle in standard position in the
Cartesian Coordinate Plane.
2. Find the trigonometric ratios of the following angles.
3. The use of a calculator as related to trigonometry could be reviewed.
Students should practise finding the values of ratios, given the measure
of an angle; and should practise finding the measure of an angle, given
the value of a trigonometric ratio.
4.
Using a calculator, find the measures of the following angles.
Program Description
Ed and Charlie bring their spacecraft, Probe 1, under control and attempt
to get it back on course. Before they can line up their flight path with
GEOSAT4 and CapCom, another angle must be calculated and Ed turns to
his "help screen."
Definitions of the three primary trigonometric ratios for an angle in
standard position are reviewed. Values of the sine, cosine, and tangent of
90° are calculated with particular attention given to the value of the tangent
ratio (undefined).
Two important relationships are discovered using an example with an
angle with its terminal arm in the second quadrant:
sin 0 = sin (180° - 0)
cos 0 = cos (180 ° - 0)
Returning to the calculations needed for Probe 1, and using a calculator, an
angle of 123.7° is found to be needed to correct the course. Ed and Charlie
are extremely happy but, once again, tragedy strikes.
Postviewing Activities
1.
Find the values of the sine, cosine, and tangent of a 180° angle. (Choose
a point on the terminal arm, calculate the value of r using the definitions.)
2.
By using diagrams similar to the one above, find the values of the
primary trigonometric ratios of 270°, 360°, and 0°. Complete the following chart.
3.
The sine, cosine, and tangent ratios have positive or negative values
depending upon the quadrants in which the terminal arms lie. For
example, in quadrant 1, the values of x, y, and r are all positive and,
therefore, the values of the primary trigonometric ratios are positive.
sin a > 0
cos a > 0
tan
a>0
Using similar diagrams for angles with terminal arms in quadrants II,
III, and IV, find the "signs" of the primary trigonometric ratios.
This information is frequently learned and recalled using what is referred to as the CAST rule. The diagram associated with this rule is:
What do the letters C, S, and T refer to? (The letter A refers to "All.")
Using the CAST rule, what are the signs of the following ratios?
sin 140°, tan 275°, cos 30 °, sin 200°.
4.
Express the following angles in the form (180° + 0), (180'- 0),
or (360° - 0) where
225°, 300°, 90°, 150°, 269°.
5. The angle a has a value between 90° and 180° (90° <_ a:5 180°). Using a
calculator and the relationships learned in this program, find the measure of a in each of the following:
6.
Using a method similar to that used in the program, prove the following relationships:
(a) sin (180° + 0) = -- sin 0
(b) cos (180 ° + 0) = -cos 0
7.
Using a calculator, find the values of sin 10° and cos 80°. Repeat this
calculation for sin 31° and cos 59 ° . It is obvious that the ratios sine and
cosine are related. Use this relationship to find what trigonometric ratio
has the same value as sin 50°.
Does this relationship hold for the other co-ratios - i.e., tangent and
cotangent, secant and cosecant?
PROGRAM FOUR.:
THE SINE LAW
Objectives
After viewing this program and completing the suggested activities and
problems, students should be able to do the following:
1. Prove the Sine Law relating the sides and angles of a triangle.
2. Recognize problems in which the Sine Law can be applied toward
reaching a solution.
3. Solve triangles using the Sine Law.
4. Solve word problems requiring the application of the Sine Law.
Previewing Activities
1. Before viewing this program, students should review the use of a
calculator to find values for trigonometric ratios. A calculator should
also be used to find the measure of an angle, given the value of a
trigonometric ratio. Practise with the following questions.
(a) Solve: sin x = 0.123
(b) In the following triangle, find a.
2. The teacher should review with the class the definitions of the trigonometric ratios for angles in standard position in a coordinate plane.
3. Some time should be spent solving right triangles and problems with
right triangles. Some examples follow.
(a) Find DE.
A
8
(c) In the given "three-dimensional" diagram, find the length of AD.
Program Description
The program opens with a review of the calculations faced by the astronauts in order to get their spacecraft back on course. A possibility of
working with oblique triangles (triangles without right angles) is introduced. It is emphasized that the methods learned in previous programs to
solve triangles are applicable to triangles with right angles only.
In order to work with oblique triangles, the Sine Law is developed in a
traditional manner. Using triangle ABC, the convention of naming sides
using lower-case letters is presented. A perpendicular is constructed from
one vertex to the opposite side (altitude), and two expressions for its length
are found and equated. The following relationship between the lengths of
sides and the sines of the angles opposite these sides is presented.
The program then returns to Ed and Charlie and their problem requiring
the calculation of an angle on the coordinate plane. An analysis of the
problem is discussed and some required calculations are made using the
Pythagorean relationship. The appropriate equality in the Sine Law is used,
and the angle which the space cadets must use is found.
Ed and Charlie continue their voyage, thankful for the Sine Law, and
eagerly anticipate more trouble and more help using the Cosine Law.
Postviewing Activities
1.
Students should learn the proof of the Sine Law and be able to prove it
for any triangle, such as triangle XYZ.
2.
Express the Sine Law in words. (In any triangle, the ratio of , . .)
3.
Use the Sine Law to find the lengths of the indicated sides in the
following diagrams:
(a) Find AB.
(b) Find ER
(c) Find RS.
(hint: find 4RTS)
D
4.
Find the measures of all "unknown" anzles in the following:
(a)
(b)
5.
E
In which of the diagrams below could you use the Sine Law to find the
length of a side or the measure of an angle?
Y
7. Solve the following problems. For each one, draw a representative
diagram as part of the solution.
(a) Two forest fire towers, E and F, are 15 km apart. How far is a third
tower G from F if angle GEF = 41° and `GFE = 58°?
(b) Given an isosceles triangle RAG, the base AG has a length of 2.3 cm.
The vertex angle R measures 50 ° . Find the lengths of the equal sides
of the triangle.
(c) A sailor sights a lighthouse on the shore and he measures
the angle between this sightline and his course to be 35°. The
sailboat travels for 2000 m and the sailor finds that the angle is
now 105°. How far is the boat from the lighthouse at the time of
the second sighting?
PROGRAM FIVE:
THE COSINE LAW
Objectives
After viewing this program, students should be able to do the following:
1. Prove the Cosine Law relating the sides and angles of a triangle.
2. Solve triangles using the Cosine Law.
Previewing Activities
1. Students should review the Sine Law and practise applying it in situations such as the following:
2. The Sine Law was developed and has been applied using triangles
which do not have a 90° angle. Determine if the Sine Law may be applied
to right-angled triangles by solving the following triangle.
Find the length of RS using:
(i) the Sine Law
(ii) the sine or cosecant ratio
3.
In preparation for the manipulation of the Cosine Law equation,
students should solve the following equations for the indicated variables.
(a) 4x - 3 = 27, solve for x.
(b) 25 = 16 + 1 + CosA; solve for CosA.
(c) 49 = 55 -10 CosB; solve for CosB.
(d) 16 = 4 + 9 -12 CosD; solve for CosD.
Program Description
A review of the Sine Law is presented at the beginning of this program. An
example, in which two angles and the contained side of a triangle are given,
is completed. A second example involving a triangle, given two sides and
the angle opposite one of these sides, is also described.
The question of working with triangles in which two sides and the contained angle are known, and in which three sides are given, is raised. The
Cosine Law is developed using traditional methods by constructing an
altitude, using the cosine ratio in one of the right-angled triangles, and
applying the Pythagorean relationship in the other triangle. After some
algebraic manipulation, the Cosine Law is obtained.
Ed and Charlie use their new relationship to calculate the remaining
distance to the centre of the galaxy. They determine that the Sine Law is not
appropriate because each equation has two "unknown quantities." The
distance, 9.4 kiloparsecs, is found.
An application of the Cosine Law, in which the lengths of three sides of a
triangle are given, is completed. The equation is solved for the cosine of one
angle, and the angle is found using a calculator. The other angles are found
using the Sine Law.
Postviewing Activities
1. Students should review the proof of the Cosine Law and attempt to
derive the formula using the following diagram:
E
3. In the following diagrams, use the Cosine Law to find the lengths of the
indicated sides.
(a) Find the length of side a.
(b) Find the length of side h.
(c) Find the length of side s.
PROGRAM SIX:
APPLICATIONS OF SINE
AND COSINE LAWS
Objectives
After viewing this program and completing subsequent suggested problems, students will be able to:
1. Determine, based on the given data, whether the Sine Law or Cosine
Law could be applied to solve triangles.
2. Apply the Sine Law and Cosine Law.
3. Solve problems in which a triangle can be constructed from the given
information. Skills related to the Pythagorean relationship, trigonometric ratios in right-angled triangles, the triangle angle-sum relationship,
and the Sine and Cosine Laws will be used.
Previewing Activities
1. Students should practise writing the Sine Law and Cosine Law statements for variously labelled triangles
(a) State the Sine Law relating the measures of the angles and the
lengths of the sides in
Program Description
CapCom is frantically attempting to contact Probe 1. Ed and Charlie report
that they are proceeding rapidly into the galactic mass, and CapCom
wishes them luck.
The Sine Law and Cosine Law are quickly recalled and the situations
(triangles with various combinations of "known" sides and angles) are
reviewed. If we are given the measure of an angle and the length of the side
opposite it, then we can apply the Sine Law. A triangle is solved to illustrate
this approach.
If we do not have this "opposite" information, such as with a triangle in
which the lengths of the three sides are known, the Cosine Law should be
applied. Practice involving the appropriate form o£ the Cosine Law equation is presented.
Ed and Charlie approach what appears to be a black hole, and by considering a triangle and applying the Cosine Law, the distance across this space
phenomenon is calculated.
The program and series draw to an end with Probe 1 entering the black hole.
Ed and Charlie experience a vast array of laser-produced shapes and colors.
Postviewing Activities
1.
Students should practise determining which law can be applied to
attack a problem involving a triangle. For example, using the "opposite" criterion, determine whether the Sine Law or Cosine Law is
appropriate to use initially in the following situations.
(b) Find side f.
2. Perform the actual calculations to determine the required values in
Question 1.
3. Solve the following problems. Remember to interpret the given data
and construct a representative diagram to illustrate this information.
Then apply the skills which you have learned.
(a) An isosceles triangle ABC has a base AB of length 10 cm. If the vertex
angle C has a measure of 52°, find the lengths of the equal sides of
the triangle.
(b) A person standing on a dock sees two sailboats out in the lake. The
two lines of sight make an angle of 31°. If the distances from the
person to the two boats are 150 m and 250 m respectively, how far
apart are the sailboats?
(c) Two trains leave the same station at the same time. Train M travels
due North at a rate of 120 km/h while train N goes South 60°West
at 100 km/h. How far apart are the trains after one hour? How far
apart are they when train N has travelled 250 km?
(d) Two planes are flying at the same altitude. At a given moment in
time, the passenger airliner is located 110 km from an airport control
tower in a direction N31° E from the tower. The second plane, a
private jet, is 40 km from the tower in a direction N4° W from the
tower. What is the distance between the planes at that moment?
(e) On a golf course, the width of the green on the eighth hole is 34.3 m.
A golf ball on the fairway is 170 m from the right side of the green
and 182 m from the left side.
Within what angle must the golfer hit the ball in order for it to land
on the green?
Ordering information
To order the videotapes or this publication, or for additional information, please
contact one of the following:
Ontario
TVOntario Sales and Licensing
Box 200, Station Q
Toronto, Ont. M4T 2T1
(416) 484-2613
United States
TVOntario
U.S. Sales Office
901 Kildaire Farm Road
Building A
Cary, North Carolina
27511
Phone: 800-331-9566
Fax: 919-380-0961
E-mail: ussale [email protected]
Programs
BPN
Trigonometric Ratios
Solving Right Triangles
Angles on the Plane
The Sine Law
The Cosine Law
Applications of Sine and Cosine Laws
298301
298302
298303
298304
298305
298306