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TRIGONOMETRIC FUNCTIONS Teacher's Guide
TRIGONOMETRICFUNCTIONS
Teacher's Guide
Trigonometric Functions 1: Solving Triangles
Teacher's Guide
The teacher's guide is designed for use with the television series TrigonometricFunctions 1: Solving Triangles. The programs are broadcast by TVOntario, thetelevision service of The Ontario Educational Communications Authority. Forbroadcast dates consult the appropriate TVOntario schedule. The programs areavailable on videotape. Ordering information for videotapes and this publicationappears on page 42.
The SeriesProducer/Director: David ChamberlainProject Officers: John Amadio
David Way
The GuideProject Leader: David ChamberlainAuthor: Ron CarrEditor: Mei-lin CheungDesigner: 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 5
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
INTRODUCTION
Trigonometry is a study of angle measurement. Rather than simple meas-urement as with the use of a protractor in plane geometry, it is calculationusing special functions which depend on angles and which are calledtrigonometric functions.
The six-program Trigonometric Functions 1: Solving Triangles defines thesefunctions using the lengths of the sides of a right-angled triangle, and usingrelationships on a coordinate plane. Using a calculator, or tables, themeasures of angles and the lengths of sides in triangles are calculated. Twospecial relationships, the Sine Law and the Cosine Law, are developed, andpractical applications related to solving triangles are presented.
Trigonometric Functions 1: Solving Triangles, as in all the series in the Con-cepts in Mathematics group, uses the contemporary 3-D computer anima-tion introduced in the first series, Vectors. The study of trigonometry is con-tinued with special reference to the properties of trigonometric graphs inthe companion series, Trigonometric Functions 2.
PROGRAM ONE:TRIGONOMETRIC RATIOS
After viewing this program and completing several suggested activitiesand problems, students should be able to do the following:
1. Locate the hypotenuse, the opposite side, and the adjacent side withrespect to a given acute angle in a right-angled triangle.
2. Use the Pythagorean relationship to calculate the length of one side ofa right-angled triangle, given the lengths of the other two sides.
3. Define the primary trigonometric ratios (sine, cosine, and tangent) foran acute angle in a right-angled triangle.
4. Use tables and calculators to find the values of the trigonometric ratiosfor a given angle.
5. Use an appropriate trigonometric ratio to find the measure of an angleor the length of a side in a right-angled triangle.
b. "Solve" right-angled triangles.
Objectives
7. Solve word problems involving right-angled triangles.
Previewing Activities
1. This program assumes that students are familiar with basic terminol-ogy related to angles and triangles. Spend some time reviewing con-cepts and definitions of the following: acute, obtuse, right-angledtriangle (right triangle), scalene, oblique, equilateral, angle-sum =180 ° .
2. Review the use of the protractor to find the measures of angles indegrees.
3. Review the Pythagorean relationship for the lengths of the sides ofright-angled triangles. A number of examples should be done usingtables of square roots and/or calculators.
4. In most cases, calculations result in answers which are correct to onedecimal place. Review this concept and discuss other methods of ex-pressing 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. Weare reintroduced to our astronaut friends, Ed and Charlie, who are incontrol of the spacecraft, and looking forward to exploring the universe.
As they proceed, the measure of an angle must be found so that theastronauts may continue to be in communication contact with MissionControl.
In order to define the primary trigonometric ratios, a right-angled triangleis introduced. The sides of this triangle are "named" with respect to a givenacute angle. The terms "hypotenuse," "opposite side," and "adjacent side"are discussed. The trigonometric ratios, sine, cosine, and tangent of an acuteangle in a right-angled triangle are defined, and appropriate notation isdiscussed.
The program shows how the measure of an acute angle is calculated whenthe lengths of the three sides are known. Trigonometric tables and the useof a calculator are introduced.
The new skills are applied to the transmission problem experienced by Edand Charlie. A triangle is created and, using trigonometry, the angleneeded to lock their signal with CapCom is calculated. Communicationcontact 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 definedas ratios of the lengths of the hypotenuse, opposite side, and adjacentside, students should practise locating these sides in right-angledtriangles. 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 becalculated using a number of different right-angled triangles andknowing the lengths of all three sides.
3. Values for the ratios should be found using right-angled triangles inwhich the lengths of two sides are given. Using the Pythagorean rela-tionship, the length of the third side is calculated first and then thevalues of the ratios are found.
4. Some time should be spent showing students that the values of thetrigonometric ratios depend on the measure of the acute angle, not on
the size of the triangle. This can be accomplished by using similar rightangled triangles and finding values of the ratios for corresponding
angles.
The use of calculators and tables should be practised. Values of trigono-metric 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
6. Students should practise finding the length of a side of a right-angledtriangle, given the measure of an angle and the length of one side. Asuggested approach is to have the students form a ratio
"unknown" sideknown side
and then determine the trigonometric ratio that this represents for thegiven angle. For example:
The equation is then solved.
ratio. For example, given sin
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°x5
x tan 38°
This may be an appropriate point, depending on the progress of theclass, to introduce the definitions of the reciprocal (secondary) trigono-metric ratios.
cosecant 0 = 1 __ hypotenusesin 0 opposite side
secant 0 = 1 _ hypotenusecos 0 _ adjacent side
cotangent 0 = 1 _ adjacent sidetan 0 - oppositeside
Find the lengths of the indicated sides in the following triangles.
8. To "solve" a triangle means to find the lengths of any "unknown" sidesand the measures of any "unknown" angles. Students should bereminded that they can use skills from trigonometry, as well as thePythagorean relationship and the angle sum of a triangle, where
A appropriate. Solve the following triangles.
9. Students should practise solving word problems involving right-angledtriangles. In each case, the student should make a rough sketch of thetriangle to illustrate the given and required information, and then drawa reasonably accurate right-angled triangle representing the question.
(a) A telephone pole is supported by a number of guy-wires which areattached to the top of the pole and to concrete footings on theground. 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 theladder from the wall? Find the angle that the ladder makes with thewall.
(c) From the observation deck of a lighthouse, the angle of depres-sion of a rowboat in the bay is 7°. If the deck is 25 m above waterlevel, how far is the boat from the lighthouse?
PROGRAM TWO:SOLVING RIGHT TRIANGLES
Objectives
After viewing this program and completing the suggested activities andproblems, students should be able to do the following:
1. Define and explain the meaning of an "angle in standard position" in aCartesian 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 distancefrom the origin to a point on the terminal arm of an angle.
3. Define the primary trigonometric ratios for an angle in standard posi-tion with point (x,y) on the terminal arm and radius r units.
4. Find the values of the three primary ratios for angles in standardposition.
5. Become aware of angles larger than 90° and angles less than 0° (directionof rotation), and use calculators to find the values of their trigonometricratios.
6. Work with problems in which two or more right-angled triangles areinvolved.
Previewing Activities
1. Before students view this program, some time should be spent review-ing some items from the previous program. The naming of the hypote-nuse, opposite and adjacent sides in a right-angled triangle should bepractised, as should the definitions of the sine, cosine, and tangentratios of an acute angle.
2. Review methods used to find the lengths of sides and the measures ofangles in right-angled triangles (Pythagoras, angle-sum =180° , trigono-metric ratios).
3. Practise finding appropriate trigonometric ratios to apply to "solve"triangles.
4. In this program the Cartesian Coordinate Plane is introduced. Studentsshould review this system including the scales on the axes, orderedpairs, and the naming of points using positive and negative realnumbers.
5. If time permits, a teacher may wish to review a method of finding thedistance between two points on the Cartesian Plane by constructing aright-angled triangle and applying the Pythagorean relationship.
Program Description
When we left Ed and Charlie in Program One, they has just solved theirproblem of how to transmit and receive messages from their home base onearth. Using trigonometry, they found that they must aim their transmis-sions at an angle of 37° to the vertical. This program opens with Mission
Control advising that, due to increased danger from meteors, the fixedtransmission satellite GEOSAT4 would be moved to a higher orbit. Ed andCharlie must move farther away from earth as well, and they realize thatnew calculations must be made.
Ed and Charlie's predicament is used as an example to review the applica-tion of trigonometric ratios to solve right-angled triangles.
The program then moves to a discussion regarding angles in the CartesianCoordinate Plane. A brief review of naming points using ordered pairs ispresented. The concept of an angle in standard position is introduced withvertex at (0,0), initial arm along the positive X-axis, and the terminal armsomewhere in the plane. An angle is defined in terms of a rotation from afixed position.
Using an example in which the point P(3,5) lies on the terminal arm, thelength of the line segment OP is found using the Pythagorean relationship.The primary trigonometric ratios, sine, cosine, and tangent of an angle instandard position, with point P(x,y) on the terminal arm and with OP = runits, are defined.
Values of trigonometric ratios for angles with terminal arms in otherquadrants are calculated.
The program ends with Ed and Charlie suitably impressed by this newinformation. They plan to continue their journey to the black hole, but in hisexcitement, Ed presses the "Don't Ever Touch" button, sending Probe 1spinning away, uncontrollably.
Postviewing Activities
1. The definitions of the primary trigonometric ratios as related to an anglein standard position must be memorized and practised. Studentsshould calculate these values for angles with terminal arms in varyingpositions.
YY
Y
2. Calculations should be made for situations in which the value of eitherx or y is not known.
3. Using an appropriate trigonometric ratio and a calculator, find themeasure 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 restric-tions. Since the value of r is always positive, the sine and cosine ratiosare defined for all real values. However, with the other four ratios, thedenominator may be zero, producing an undefined value for the ratio.
5. Students may be interested in why we seem to always rotate in acounterclockwise direction. It should be explained that, from its initialposition on the positive X-axis, the terminal arm may rotate in eitherdirection. By convention, mathematicians have decided that, if therotation is counterclockwise, the angle will be considered positive. Ifthe direction of rotation is clockwise, the angle is considered to benegative.
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 co-terminal angles. For example, 45° and -315° are co-terminal angles.
Find three angles which are co-terminal with 75 ° . (Remember that youcan rotate clockwise and counterclockwise, and that you can rotatemore than one revolution.)
How many angles are co-terminal with 75°?
7. Some problems requiring the application of trigonometry involve morethan 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 andproblems, 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 relationshipssin (180° - 0) = sin 0cos (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 inquadrants II, III, and IV.
5. Find the primary trigonometric ratios of an angle with its terminal armon a coordinate axis (0°, 90 ° ,180°, 270 ° , 360°).
Previewing Activities
1. Before viewing this program, students should review the definitions ofthe primary trigonometric ratios for an angle in standard position in theCartesian 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 measureof an angle; and should practise finding the measure of an angle, giventhe 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 attemptto get it back on course. Before they can line up their flight path withGEOSAT4 and CapCom, another angle must be calculated and Ed turns tohis "help screen."
Definitions of the three primary trigonometric ratios for an angle instandard position are reviewed. Values of the sine, cosine, and tangent of90° are calculated with particular attention given to the value of the tangentratio (undefined).
Two important relationships are discovered using an example with anangle 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, anangle of 123.7° is found to be needed to correct the course. Ed and Charlieare extremely happy but, once again, tragedy strikes.
Postviewing Activities
1. Find the values of the sine, cosine, and tangent of a 180° angle. (Choosea point on the terminal arm, calculate the value of r using the defini-tions.)
2. By using diagrams similar to the one above, find the values of theprimary trigonometric ratios of 270°, 360°, and 0°. Complete the follow-ing chart.
3. The sine, cosine, and tangent ratios have positive or negative valuesdepending upon the quadrants in which the terminal arms lie. Forexample, 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 re-ferred 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),
225°, 300°, 90°, 150°, 269°.
5. The angle a has a value between 90° and 180° (90° <_ a:5 180°). Using acalculator and the relationships learned in this program, find the meas-ure of a in each of the following:
6. Using a method similar to that used in the program, prove the follow-ing 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 thiscalculation for sin 31° and cos 59 ° . It is obvious that the ratios sine andcosine are related. Use this relationship to find what trigonometric ratiohas the same value as sin 50°.
Does this relationship hold for the other co-ratios - i.e., tangent andcotangent, secant and cosecant?
or (360° - 0) where
PROGRAM FOUR.:THE SINE LAW
Objectives
After viewing this program and completing the suggested activities andproblems, 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 towardreaching 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 acalculator to find values for trigonometric ratios. A calculator shouldalso be used to find the measure of an angle, given the value of atrigonometric 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 trigono-metric ratios for angles in standard position in a coordinate plane.
3. Some time should be spent solving right triangles and problems withright 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 astro-nauts in order to get their spacecraft back on course. A possibility ofworking with oblique triangles (triangles without right angles) is intro-duced. It is emphasized that the methods learned in previous programs tosolve triangles are applicable to triangles with right angles only.
In order to work with oblique triangles, the Sine Law is developed in atraditional manner. Using triangle ABC, the convention of naming sidesusing lower-case letters is presented. A perpendicular is constructed fromone vertex to the opposite side (altitude), and two expressions for its lengthare found and equated. The following relationship between the lengths ofsides and the sines of the angles opposite these sides is presented.
The program then returns to Ed and Charlie and their problem requiringthe calculation of an angle on the coordinate plane. An analysis of theproblem is discussed and some required calculations are made using thePythagorean 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, andeagerly 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 itfor 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 thefollowing diagrams:
(a) Find AB.
(b) Find ER D
(c) Find RS.(hint: find 4RTS)
4. Find the measures of all "unknown" anzles in the following:
E
5. In which of the diagrams below could you use the Sine Law to find thelength of a side or the measure of an angle?
(b)
(a)
Y
7. Solve the following problems. For each one, draw a representativediagram as part of the solution.
(a) Two forest fire towers, E and F, are 15 km apart. How far is a thirdtower 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 sidesof the triangle.
(c) A sailor sights a lighthouse on the shore and he measuresthe angle between this sightline and his course to be 35°. Thesailboat travels for 2000 m and the sailor finds that the angle isnow 105°. How far is the boat from the lighthouse at the time ofthe 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 situ-ations such as the following:
2. The Sine Law was developed and has been applied using triangleswhich do not have a 90° angle. Determine if the Sine Law may be appliedto 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 vari-ables.
(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. Anexample, in which two angles and the contained side of a triangle are given,is completed. A second example involving a triangle, given two sides andthe angle opposite one of these sides, is also described.
The question of working with triangles in which two sides and the con-tained angle are known, and in which three sides are given, is raised. TheCosine Law is developed using traditional methods by constructing analtitude, using the cosine ratio in one of the right-angled triangles, andapplying the Pythagorean relationship in the other triangle. After somealgebraic manipulation, the Cosine Law is obtained.
Ed and Charlie use their new relationship to calculate the remainingdistance to the centre of the galaxy. They determine that the Sine Law is notappropriate because each equation has two "unknown quantities." Thedistance, 9.4 kiloparsecs, is found.
An application of the Cosine Law, in which the lengths of three sides of atriangle are given, is completed. The equation is solved for the cosine of oneangle, and the angle is found using a calculator. The other angles are foundusing the Sine Law.
Postviewing Activities
1. Students should review the proof of the Cosine Law and attempt toderive the formula using the following diagram:
E
3. In the following diagrams, use the Cosine Law to find the lengths of theindicated 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 LAWSObjectives
After viewing this program and completing subsequent suggested prob-lems, students will be able to:
1. Determine, based on the given data, whether the Sine Law or CosineLaw 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 giveninformation. Skills related to the Pythagorean relationship, trigonomet-ric 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 state-ments for variously labelled triangles
(a) State the Sine Law relating the measures of the angles and thelengths of the sides in
Program Description
CapCom is frantically attempting to contact Probe 1. Ed and Charlie reportthat they are proceeding rapidly into the galactic mass, and CapComwishes them luck.
The Sine Law and Cosine Law are quickly recalled and the situations(triangles with various combinations of "known" sides and angles) arereviewed. If we are given the measure of an angle and the length of the sideopposite 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 inwhich the lengths of the three sides are known, the Cosine Law should beapplied. Practice involving the appropriate form o£ the Cosine Law equa-tion is presented.
Ed and Charlie approach what appears to be a black hole, and by consid-ering a triangle and applying the Cosine Law, the distance across this spacephenomenon 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 toattack a problem involving a triangle. For example, using the "oppo-site" criterion, determine whether the Sine Law or Cosine Law isappropriate to use initially in the following situations.
(b) Find side f.
2. Perform the actual calculations to determine the required values inQuestion 1.
3. Solve the following problems. Remember to interpret the given dataand 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 vertexangle C has a measure of 52°, find the lengths of the equal sides ofthe triangle.
(b) A person standing on a dock sees two sailboats out in the lake. Thetwo lines of sight make an angle of 31°. If the distances from theperson to the two boats are 150 m and 250 m respectively, how farapart are the sailboats?
(c) Two trains leave the same station at the same time. Train M travelsdue North at a rate of 120 km/h while train N goes South 60°Westat 100 km/h. How far apart are the trains after one hour? How farapart are they when train N has travelled 250 km?
(d) Two planes are flying at the same altitude. At a given moment intime, the passenger airliner is located 110 km from an airport controltower in a direction N31° E from the tower. The second plane, aprivate jet, is 40 km from the tower in a direction N4° W from thetower. 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 greenand 182 m from the left side.Within what angle must the golfer hit the ball in order for it to landon the green?
Ordering information
To order the videotapes or this publication, or for additional information, pleasecontact one of the following:
OntarioTVOntario Sales and LicensingBox 200, Station QToronto, Ont. M4T 2T1(416) 484-2613
United States
TVOntarioU.S. Sales Office901 Kildaire Farm RoadBuilding ACary, North Carolina27511Phone: 800-331-9566Fax: 919-380-0961E-mail: [email protected]
Programs BPN
Trigonometric Ratios 298301Solving Right Triangles 298302Angles on the Plane 298303The Sine Law 298304The Cosine Law 298305Applications of Sine and Cosine Laws 298306
