Mathematics 20-1



MATHEMATICS 20-1TrigonometryHigh School collaborative venture withEdmonton Christian, Harry Ainlay, J. Percy Page, Jasper Place, Millwoods Christian, Ross Sheppard and W. P. WagnerEdm Christian High: Aaron TrimbleHarry Ainlay: Ben LuchkowHarry Ainlay: Darwin HoltHarry Ainlay: Lareina RezewskiHarry Ainlay: Mike ShrimptonJ. Percy Page: Debbie YoungerJasper Place: Matt KatesJasper Place: Sue DvorackMillwoods Christian: Patrick YpmaRoss Sheppard: Patricia ElderW. P. Wagner: Amber SteinhauerFacilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)2010 - 2011TABLE OF CONTENTSSTAGE 1 DESIRED RESULTSPAGEBig IdeaEnduring UnderstandingsEssential Questions444Knowledge Skills56STAGE 2 ASSESSMENT EVIDENCETransfer Task (on a separate page which could be photocopied & handed out to students)“Tri” AlbertaTeacher Notes for Transfer Task and RubricTransfer Task and RubricRubricPossible Solution891214STAGE 3 LEARNING PLANSLesson #1 Angles in Standard Position18Lesson #2 Reference Triangles & Trigonometry Ratios for Angles 0? - 360?24Lesson #3 Applying the CAST Rule29Lesson #4 Special Angles 0-30-45-60-9033Lesson #5 The Sine Law38Lesson #6 The Cosine Law 42Lesson #7 The Ambiguous Case45 Mathematics 20-1 TrigonometrySTAGE 1 Desired Results Big Idea: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus.Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often. Enduring Understandings:Students will understand …Angles in a circle can be expressed in a variety of ways. Primary trigonometric ratios and Pythagorean Theorem only work for right triangles, while the sine and cosine law will work for all triangles.Each primary trigonometric ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST).Special triangles are useful for determining exact value of trigonometric ratios with reference angles 0, 30o, 45o and 60o. Essential Questions:What is triangulation?How is a negative angle possible?How many different ways can you estimate the height of a mountain?Implementation note: Ask students to consider one of the essential questions every lesson or two.Has their thinking changed or evolved?Why is it easier to find the exact value of some trigonometric ratios? Knowledge:Enduring UnderstandingList enduring understandings (the fewer the better)Specific OutcomesList the reference # from the Alberta Program of StudiesDescription ofKnowledgeThe paraphrased outcome that the group is targetingStudents will understand…Angles in a circle can be expressed in a variety of ways. *T1, T2, T3Students will know …an angle in standard position, given the measure of the anglewithout the use of technology, the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ, where θ = 0?, 90?, 180?, 270? or 360?the sign of a given trigonometric ratio for a given angle, without the use of technology, and explainthe patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°Students will understand…Primary trigonometric ratios and Pythagorean Theorem only work for right triangles, while the sine and cosine law will work for all triangles. T2, T3Students will know …the Pythagorean theorem or the distance formula, can be used to calculate the distance from the origin to a point P (x, y) on the terminal arm of an angle some contextual problems can be solved using trigonometric ratiosStudents will understand…Each primary trigonometry ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST). T1, T2Students will know …illustrate, using examples, that the points P ( x, y), P (?x, y), P (?x,? y) and P (x,? y) are points on the terminal sides of angles in standard position that have the same reference angle Students will understand…Special triangles are useful for determining exact value of trigonometric ratios with reference angles 0, 30o, 45o and 60o. T1, T2Students will know …determine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?describe patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°8888I*T = Trigonometry Skills: Enduring UnderstandingList enduring understandings (the fewer the better)Specific OutcomesList the reference # from the Alberta Program of StudiesDescription of SkillsThe paraphrased outcome that the group is targetingStudents will understand…Angles in a circle can be expressed in a variety of ways. T1Students will be able to…sketch an angle in standard position, given the measure of the angledetermine the reference angle for an angle in standard positionexplain, using examples, how to determine the angles from 0° to 360° that have the same reference angle as a given angleillustrate, using examples, that any angle from 90° to 360° is the reflection in the x-axis and/or the y-axis of its reference angledetermine the quadrant in which a given angle in standard position terminatesdraw an angle in standard position given any point P (x, y) on the terminal arm of the angleStudents will understand…Primary trigonometric ratios and Pythagorean Theorem only work for right triangles, while the Sine and cosine law will work for all triangles. T2,3Students will be able to…determine, using the Pythagorean theorem or the distance formula, the distance from the origin to a point P (x, y) on the terminal arm of an anglesolve a contextual problem, using trigonometric ratiossketch a diagram to represent a problem that involves a triangle without a right anglesolve, using primary trigonometric ratios, a triangle that is not a right triangleexplain the steps in a given proof of the sine law or cosine lawsketch a diagram and solve a problem, using the cosine lawsketch a diagram and solve a problem, using the sine lawStudents will understand…Each ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST). T1, T2Students will be able to…illustrate, using examples, that the points P (x, y), P (?x, y), P (?x,? y) and P (x,? y) are points on the terminal sides of angles in standard position that have the same reference angledetermine the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θdetermine the sign of a given trigonometric ratio for a given angle, without the use of technology, and explainsolve, for all values of θ, an equation of the form sin θ = a or cos θ = a, where ?1 ≤ a ≤ 1, and an equation of the form tan θ = a, where a is a real numberdetermine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?describe patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°Students will understand…Special triangles are useful for determining exact value of trigonometric ratios with reference angles 0, 30o, 40o and 60o T2Students will be able to…determine, without the use of technology, the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ, where θ = 0?, 90?, 180?, 270? or 360?determine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?Implementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.*T = TrigonometrySTAGE 2 Assessment Evidence1 Desired Results Desired ResultsDesired Results Desired Results “Tri” Alberta Teacher NotesThere is one transfer task to evaluate student understanding of the concepts in relation to Trigonometry. A photocopy-ready version of the transfer task is included in this section.Implementation note:Students must be given the transfer task & rubric at the beginning of the unit. They need to know how they will be assessed and what they are working toward.Each student will:Estimate the distances and angles between four chosen locations, where Edmonton is the origin. The estimations will be verified using the sine law and cosine law. Solve for reference, principle and, co-terminal angles, and approximate coordinates of each location.Materials:rulerprotractormap – provided in transfer taskInternetYou may wish to assign Part A as a take home assignment. This will eliminate the need for Internet access. “Tri” Alberta - Student Assessment TaskYou have access to your own helicopter to visit four locations over two days. You can only visit two locations per day and for each excursion, you must visit locations in adjacent quadrants. You are travelling back to Edmonton at the end of each day. Your two-day trip must include all 4 quadrants.Part A: Background InformationChoose four locations in Alberta, each in a different quadrant. You must research two important facts to explain why you are choosing these locations. (for example: Vegreville has a giant Ukrainian egg (pysanka). Draw two triangles of your excursions, starting and ending in Edmonton. (Each triangle must contain two locations in adjacent quadrants and Edmonton.)Part B: Excursion OneMeasure all three sides of one triangle using a ruler. Use the map scale to determine the actual distances. (Round the distances to the nearest km.) Determine the measure of all three angles using the cosine law. (Round your answers to the nearest degree.)Verify each angle using the sine law. Part C:Excursion TwoMeasure the distances from Edmonton to the other two locations in the second triangle. Using a ruler convert the measured distances to the actual distances using the map scale.Measure the angle at Edmonton, in between those two distances, using a protractor to the nearest degree.Determine the distance between the two non-Edmonton locations using the cosine law.Determine the other two angles using the sine law.Verify algebraically the measured angle and distances of the previous bullets. Verify the angle using the sine law, and sides using the cosine law. “Tri” Alberta - Student Assessment TaskPart D:Reference AnglesName and measure the reference angles for each one of your locations (Use a protractor to the nearest degree.)Name and determine both the principle, and negative co-terminal angles for each location.Provide an approximate coordinate for each location. (Use the scale on the map and round to the nearest km.)Glossaryadjacent quadrants – Two quadrants beside each otherambiguous case – From the given information the solution for the triangle is not clear: there might be one triangle, two triangles, or no triangle [Math 20-1 (McGraw-Hill Ryerson: page 104)]angle in standard position - The location of an angle in the plane in which the vertex is at the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotatecosine law - the relationship between the cosine of an angle and the lengths of the three sides of any trianglec2 = a2 + b2 = 2ab cos Ccoterminal angle – An angle in standard position with the same initial arm and terminal arm as the principal angle. Adding or subtracting the principal angle by a multiple of 360° finds coterminal angles.exact value – Answers involving radicals or fractions are exact, unlike approximated decimal values [Math 20-1 (McGraw-Hill Ryerson: page 587)]oblique triangle – A triangle that is not a right triangleprinciple angle - The smallest positive anglequadrantal angle – an angle in standard position where the terminal arm is on the x- or y-axis. Examples are 0°, 90°, 180°, 270° and 360°.reference angle – The acute angle formed by the terminal arm of an angle in standard position and the x-axisrotation angles:positive angle - An angle in standard position swept out by a counterclockwise rotation of its terminal armnegative angle - An angle in standard position swept out by a clockwise rotation of its terminal armreference triangle – A right triangle with a reference angle as one of its verticessine law - The lengths of the sides are proportional to the sines of the opposite angles AssessmentMathematics 20-1Trigonometry - Rubric Level CriteriaExcellent4Proficient3Adequate2Limited*1Insufficient / Blank*Performs Algebraic Operations and Verification using Sine Law (Parts B and C)Student is able to determine and verify all angles and sides in both oblique triangles Student is able to determine and verify four or five out of six angles and sides in both oblique triangles Student is able to determine and verify three out of six angles and sides in both or either oblique triangles Student is able to determine and/or verify one or two angles or sides in either or both oblique triangles Student is unable to determine or verify any angles or sides in either oblique triangle Performs Algebraic Operations and Verification using Cosine Law (Parts B and C)Student is able to determine and verify all angles and sides in both oblique trianglesStudent is able to determine and verify four or five out of six angles and sides in both oblique trianglesStudent is able to determine and verify three out of six angles and sides in both or either oblique trianglesStudent is able to determine and/or verify one or two angles or sides in either or both oblique trianglesStudent is unable to determine or verify any angles or sides in either oblique triangleSolving for Reference, Principle, Negative Co-terminal Angles and Coordinates(Part D)Student is able to solve all angles and coordinates for each locationStudent is able to solve twelve out of sixteen angles and coordinates for each locationStudent is able to solve eight out of sixteen angles and coordinates for each locationStudent is able to solve four out of sixteen angles and coordinates for each locationStudent is unable to solve any angles or coordinates for each locationPresentation (Parts A – D)Student has presented all clear and accurate diagrams solutions, andProvides relevant reasons for locations chosenStudent has presented most clear and accurate diagrams, solutions, and provides relevant reasons for locations chosen. Student has presented some clear and accurate diagrams, solutions, and provides relevant reasons for locations chosen.Student has presented no clear and accurate diagrams, solutions, and fails to provide relevant reasons for locations chosenStudent has not presented.Possible Solution to “Tri” AlbertaAnswers will vary depending on the locations students chose.The following solutions are from Edmonton to Fort Chipewyan to Spirit River for excursion one and Edmonton to Crowsnest Pass to Oyen for excursion two.Part A:B.Fort Chipewyan- oldest community in Alberta- population 1012C.Spirit River - "Chepi Sepe" - Cree for Ghost or Spirit River- Crowsnest Pass – Frank Slide covered the city- "Burmis Tree" – 700 year old tree died and fell but was re-builtE.Oyen - 1908 Andrew Oyen walked from Spokane, Washington to Oyen- Canada’s National Women’s Hockey team coach (2 olympic gold medals) is from OyenPart B:Measured Distances:Edmonton (A) to Fort Chipewyan (B) = 9.8 cm x 60 = 588 kmEdmonton ( A) to Spirit River (C) = 6.7 x 60 = 402 kmFort Chipewyan (B) to Spirit River (C) = 9.3 x 60 = 558 kmSolved Angles:cos A = A = 65.5°cos B = B = 41.0° cos C = C = 73.5°Verify:A = 65.5°B = 41.0° C = 73.5°Part C:Measured Distances:DAE = 53.0°Edmonton (A) to Crowsnest Pass (D) = 7.2 cm x 60 = 432 kmEdmonton (A) to Owen (E) = 5.3 x 60 = 318 kmCalculations(DE)2 = 3182 + 4322 – 2 x 318 x 432 x cos 53.0oDE = 349.9 kmD = 46.5°E = 80.4°Verify:(AD)2 = 3182 + 349.92 – 2 x 318 x 349.9 x cos 80.4oDE = 431.8 km(AE)2 = 4322 + 349.92 – 2 x 432 x 349.9 x cos 46.5oAE = 317.7 kmPart D:<BAQreference angle = 77o principle angle = 77 o coterminal angle = -283oB = (126, 576) <PACreference angle = 38o principle angle = 142o coterminal angle = -218oC = (-324, 246)<PADreference angle = 81o principle angle = 261 o coterminal angle = -99 oD = (-66, -426) <QAEreference angle = 46o principle angle = 314 o coterminal angle = -46oE = (222, -228)STAGE 3 Learning PlansLesson 1Angles in Standard PositionSTAGE 1BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus.ENDURING UNDERSTANDINGS: Students will understand …Angles in a circle can be expressed in a variety of ways. Special triangles are useful for determining exact value of trigonometric ratios with reference angles 0o, 30o, 40o and 60o.ESSENTIAL QUESTIONS: What is triangulation?How is a negative angle possible?How many different ways can you estimate the height of a mountain?Why is it easier to find the exact value of some trigonometric ratios?KNOWLEDGE: Students will know …an angle in standard position, given the measure of the anglethe reference angle for an angle in standard positionthe exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?SKILLS: Students will be able to …sketch an angle in standard position, given the measure of the angledetermine the reference angle for an angle in standard positionexplain, using examples, how to determine the angles from 0° to 360° that have the same reference angle as a given angleillustrate, using examples, that any angle from 90° to 360° is the reflection in the x-axis and/or the y-axis of its reference angledetermine the quadrant in which a given angle in standard position terminatesdraw an angle in standard position given any point P (x, y) on the terminal arm of the angleillustrate, using examples, that the points P ( x, y), P (?x, y), P (?x,? y) and P (x,? y) are points on the terminal sides of angles in standard position that have the same reference angleImplementation note:Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.Lesson SummaryDemonstrate/explore angles in standard position. Lesson PlanHookShow a video clip of someone doing a three-sixty (360? turn). Video 1: 2: 1Video 2files were added to the EPSB Understanding by Design share siteAsk them to estimate how many degrees the person spun. Have students try to do a 90?, a 180?, a 270? turn (others if you deem it important to do so. Ask which way they spun (clockwise/counter-clockwise). Did the people who went in opposite directions create the same angle? Discuss.Lesson GoalStudents will demonstrate an understanding of angles in standard position (0? - 360?).Activate Prior KnowledgeDiscuss the idea of 360? as a circle (skateboarding, snowboarding etc.)Remind students about acute/right/obtuse/straight/reflex anglesReview the concept of Cartesian plane, numbering of quadrants and how to locate points on the plane.LessonDefine the parts of an angle (initial arm, terminal arm, vertex, rotation angle, standard position).Have students draw a coordinate plane, and draw lines estimating multiples of 30? and 45?. Check with a protractor if required.Define reference angle, find a reference angle in all of the examples students have already done.Use this applet to show the reflections of the angles in the other 3 quadrants. Talk about the colours of the initial arm vs. the terminal arm (blue vs. red).source: 1: Give students a reference angle and a quadrant and have them tell you the rotation angle. Option 2: Give a rotation angle and ask for the reference angle and quadrant number. Example: Reference QuadrantRotation 30?4210?67?2130?Use examples to illustrate that the points P (x, y), P (?x, y), P (?x, ?y) and P (x, ?y) are points on the terminal sides of angles in standard position that have the same reference angle. Give the students the point (6, 8) for P (x, y), and asks them to write down the reflected points in quadrants II, III and IV:P (?x, y)P (?x, ?y)P (x, ?y).Briefly discuss the connection between the original point given and a triangle. From the original point on the terminal arm (and the reflected points in other 3 quadrants) draw a vertical line to the x-axis. Question: What do you notice about the 4 triangles? Answer: They are congruent and are right-angled. Refer back to the applet. source: next lesson will focus on the right-angled triangles that we have created. The angles that determine the height of these triangles are called reference angles. Going BeyondDiscuss or have students research the concepts of negative angles versus positive angles, co-terminal angles, principal angles etc. 20-1 (McGraw-Hill Ryerson: sec 2.1, pages 74-87)Ron Blond’s Trig Applet SupportingAssessmentOption 1: Exit slip showing rotation angle, quadrant, reference angle. Have students fill in the blanks.Option 2: Exit slip “It is 3:15 pm, if you rewind your clock and the minute hand rotates 120?, what time is it?”Answer: 11:55 am Glossaryangle in standard position - The location of an angle in the plane in which the vertex is at the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotateexact value – Answers involving radicals or fractions are exact, unlike approximated decimal values [Math 20-1 (McGraw-Hill Ryerson: page 587)]initial arm - For an angle in standard position, the arm along the positive x-axisreference angle – The acute angle formed by the terminal arm of an angle in standard position and the x-axisrotation angles:positive angle - An angle in standard position swept out by a counterclockwise rotation of its terminal armnegative angle - An angle in standard position swept out by a clockwise rotation of its terminal armterminal arm – For an angle in standard position, the arm that is free to rotatevertex - Common endpoint of two rays that form the angle.Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.OtherLesson 2Reference Triangles & Trigonometry Ratios for Angles 0? - 360?STAGE 1BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus.ENDURING UNDERSTANDINGS: Students will understand …Angles in a circle can be expressed in a variety of ways. Each primary trigonometric ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST).Special triangles are useful for determining exact value of trigonometric ratios with reference angles 0o, 30o, 40o and 60o.ESSENTIAL QUESTIONS: What is triangulation?How many different ways can you estimate the height of a mountain?Why is it easier to find the exact value of some trigonometric ratios?KNOWLEDGE: Students will know …an angle in standard position, given the measure of the anglethe reference angle for an angle in standard positionthe Pythagorean theorem or the distance formula, can be used to calculate the distance from the origin to a point P (x, y) on the terminal arm of an anglethe sign of a given trigonometric ratio for a given angle, without the use of technology, and explainthe exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?the patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°some contextual problems can be solved using trigonometric ratiosSKILLS: Students will be able to …determine, using the Pythagorean theorem or the distance formula, the distance from the origin to a point P (x, y) on the terminal arm of an angledetermine the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ.determine, without the use of technology, the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ, where θ = 0?, 90?, 180?, 270? or 360?determine the sign of a given trigonometric ratio for a given angle, without the use of technology, and explainsolve, for all values of θ, an equation of the form sin θ = a or cos θ = a, where ?1 ≤ a ≤ 1, and an equation of the form tan θ = a, where a is a real numberdetermine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?describe patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°solve a contextual problem, using trigonometric ratiosLesson SummaryRelate the 3 primary trigonometric ratios to angles in standard position.Determine the sign of a given trig ratio for a given angle, without the use of technology and explain.Describe patterns in and among the values of sine, cosine, and tangent ratios for angles from 0? - 360?. Lesson PlanHook – Look at the school flagpole. How can we measure to the top? Go out to the pole with a clinometer and a tape measure.Lesson GoalRelate SOH CAH TOA and Pythagorean theorem to reference angles.Activate Prior KnowledgeReview SOH CAH TOA and Pythagorean Theorem using a right triangle.Define the primary trigonometric ratios and the sides of a triangle.LessonRevisit the triangle and its congruent reflections from the previous day where P (x, y) = P (6, 8) Define: Reference Triangle.Take each triangle separately and use the Pythagorean theorem to find the distance from point P to the origin. Does the distance vary in the other 3 quadrants? Show the primary trigonometric ratios in conjunction with the terminal arm in each quadrant. Q: What do we know about the reference triangles? A: They are all congruent.Have students calculate the reference angle and the rotation angles. Continue using the 4 quadrants and ask “Where is cosine positive? Where is sine positive? Where is tan positive?” Define the CAST rule.You may want to return to Ron Blond’s applet at this point and look at it again. source: students if the given trig ratios will be positive or negative.tan 217?cos 122?sin 300?cos 50?In which quadrant is each of the following located?Revisit the CAST rule by looking at the results of the above activity.As a demonstration, draw a Cartesian plane, choose a point in quadrant I, and draw a terminal arm through the point. Determine:distance from the origin to the pointthe exact value of the sin θ, cos θ, and tan θthe reference angleStudent Pairs Activity: Students pick any point in quadrant I and draw a terminal arm. Students should then swap papers and determine:distance from the origin to the pointthe exact value of the sin θ, cos θ, and tan θthe reference angle Going BeyondContinue the above lesson using negative angles and angles beyond 360ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 2.2, pages 88-99)source: Supporting Assessment Glossaryangle in standard position - The location of an angle in the plane in which the vertex is at the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotateinitial arm - For an angle in standard position, the arm along the positive x-axisquadrantal angle – an angle in standard position where the terminal arm is on the x- or y-axis. Examples are 0°, 90°, 180°, 270° and 360°.reference angle – The acute angle formed by the terminal arm of an angle in standard position and the x-axisrotation angles:positive angle - An angle in standard position swept out by a counterclockwise rotation of its terminal armnegative angle - An angle in standard position swept out by a clockwise rotation of its terminal armreference triangle – A right triangle with a reference angle as one of its verticesterminal arm – For an angle in standard position, the arm that is free to rotateGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.vertex - Common endpoint of two rays that form the angle.OtherLesson 3Applying the CAST RuleSTAGE 1BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus.ENDURING UNDERSTANDINGS: Students will understand …Angles in a circle can be expressed in a variety of ways. Each primary trigonometric ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST).ESSENTIAL QUESTIONS: What is triangulation?How many different ways can you estimate the height of a mountain?Why is it easier to find the exact value of some trigonometric ratios?KNOWLEDGE: Students will know …the sign of a given trigonometric ratio for a given angle, without the use of technology, and explainthe exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?the patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°some contextual problems can be solved using trigonometric ratiosSKILLS: Students will be able to …determine the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θdetermine, without the use of technology, the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ, where θ = 0?, 90?, 180?, 270? or 360?determine the sign of a given trigonometric ratio for a given angle, without the use of technology, and explainsolve, for all values of θ, an equation of the form sin θ = a or cos θ = a, where ?1 ≤ a ≤ 1, and an equation of the form tan θ = a, where a is a real numberdetermine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?describe patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°solve a contextual problem, using trigonometric ratiosLesson SummaryReview the CAST ruleSolve an equation of the form sin θ = a or cos θ = a, for all values of θ, where , and tan θ = a, where a is real. Lesson Plan - Applying the CAST RuleHookAsk students for “real-life” uses of trigonometry (navigation, construction industry, astronomy, space exploration, design etc.). Make a list. Then show this video for fun, which shows 5 ways that trigonometry can be used in your everyday life.Video 3: (apologies to non-Gaga fans)Video 3file was added to the EPSB Understanding by Design share siteLesson GoalUse the CAST rule and reference triangles to solve trig equations where the angle is between 0? and 360?.Activate Prior KnowledgeReview the CAST rule and reference triangles. May want to use Ron Blond’s applet again here.source: : Make a large coordinate plane using masking tape on the floor. Have students stand on the outside of the grid.Teacher says: “Tan Negative”, or “Sin Positive”, etc. and students must run to a quadrant where that is correct.Draw the coordinate plane on the board, ask the students to put CAST in the correct quadrants. Ask them the meaning of it.Review reference angles and triangles by giving them this question or one like it:Examples:Ask if θ = 140o, what is the reference angle? Draw the reference triangle and find all other angles that have the same reference angle. Use your calculator to find sin θ, cos θ, and tan θ of every one of those angles in all four quadrants.If , a) Find cos θ and tan θ as exact values.b) Find all possible values of θ, where θ is between 0? and 360?. If , a) Find cos θ and tan θ as exact values.b) Find all possible values of θ, where θ is between 0? and 360?.If , a) Find cos θ and tan θ as exact values.b) Find all possible values of θ, where θ is between 0 and 360?.Point P (-1, -8) is on the terminal arm of an angle. Find the angle.If , Find all possible values of θ, where θ is between 0? and 360?. Round to the nearest degree. Going BeyondSolve equations like and etc. (You may want to teach the quadratics unit first, so students are familiar with how to solve.)ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 2.2) Ways to use Trigonometry in Everyday Life Video 3: 3file was added to the EPSB Understanding by Design share site Supporting AssessmentExit Slip: If , find all possible values of θ, where θ is between 0 and 360 degrees. Round to the nearest degree. Glossaryangle in standard position - The location of an angle in the plane in which the vertex is at the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotateinitial arm - For an angle in standard position, the arm along the positive x-axisquadrantal angle – an angle in standard position where the terminal arm is on the x- or y-axis. Examples are 0°, 90°, 180°, 270° and 360°.reference angle – The acute angle formed by the terminal arm of an angle in standard position and the x-axisrotation angles:positive angle - An angle in standard position swept out by a counterclockwise rotation of its terminal armnegative angle - An angle in standard position swept out by a clockwise rotation of its terminal armreference triangle – A right triangle with a reference angle as one of its verticesterminal arm – For an angle in standard position, the arm that is free to rotateGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.vertex - Common endpoint of two rays that form the angle.OtherLesson 4Special Angles 0-30-45-60-90STAGE 1BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus.ENDURING UNDERSTANDINGS: Students will understand …Angles in a circle can be expressed in a variety of ways. Each primary trigonometric ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST).Special triangles are useful for determining exact value of trigonometric ratios with reference angles 0o, 30o, 40o and 60o.ESSENTIAL QUESTIONS: What is triangulation?How is a negative angle possible?How many different ways can you estimate the height of a mountain?Why is it easier to find the exact value of some trigonometric ratios?KNOWLEDGE: Students will know …the reference angle for an angle in standard positionthe Pythagorean theorem or the distance formula, can be used to calculate the distance from the origin to a point P (x, y) on the terminal arm of an anglethe sign of a given trigonometric ratio for a given angle, without the use of technology, and explainthe exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?the patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°some contextual problems can be solved using trigonometric ratiosSKILLS: Students will be able to …determine, using the Pythagorean theorem or the distance formula, the distance from the origin to a point P (x, y) on the terminal arm of an angledetermine the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θdetermine, without the use of technology, the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ, where θ = 0?, 90?, 180?, 270? or 360?determine the sign of a given trigonometric ratio for a given angle, without the use of technology, and explainsolve, for all values of θ, an equation of the form sin θ = a or cos θ = a, where ?1 ≤ a ≤ 1, and an equation of the form tan θ = a, where a is a real numberdetermine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?describe patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360°solve a contextual problem, using trigonometric ratiosLesson SummaryDetermine the exact value of sine, cosine, or tangent with a given angle, with a reference angle of 0?, 30?, 45?, 60? or 90?.Determine all possible angles between 0? and 360?, without the use of technology, given an exact trigonometric ratio. Lesson PlanHookSee .../sculpturegarden/sculpture/sculpture12.shtm link to structure/sculpture built entirely of equilateral and isosceles triangles. It’s famous.Lesson GoalTo determine the exact value of sine, cosine, or tangent with a given angle, with a reference angle of 30?, 45? or 60? and determine all possible angles between 0? and 360?, without the use of technology, given an exact trigonometric ratio.Activate Prior KnowledgeReview previous day’s homework.LessonDraw a random square on the page (big or small). Tell students to decide how long the side is (1 unit, 2 units, 3 units etc.) Draw a diagonal and ask how many degrees the other 2 angles are. Given your chosen lengths, find the length of the diagonal. Find the 3 primary trigonometric ratios for the angle. Record the answers from 3 students and draw their diagrams on the board and students will see that similar triangles were created. The answers are the same no matter what size of square they started with. (Note: Hopefully you are teaching this AFTER you have taught Rational Expressions, since they will have to rationalise the denominator in order to see that the trig ratios are equal.)112233Although any square will work, since all of the other triangles drawn are similar triangles, discuss why it is easiest to work with the unit square. Confirm the answers for the trigonometry ratios using the calculator. Discuss what an exact value of a trigonometry ratio is vs a rounded value.Next, start with an equilateral triangle with sides of 2 units, cut it in half to make 2 right triangles (in order to use SOH CAH TOA). One of the angles becomes 30?. Find the sine, cosine and tangent ratios of the 30? and 60? angles.Ask the students if it would make a difference if our original isosceles triangle only had one-unit lengths.Those steps should lead to these 2 special triangles. Memorise these triangles.Examples:Using the CAST rule and these special triangles, find:sin 240°Suggested steps:Draw the reference triangle on the coordinate plane and find the reference angle.Have the students decide whether the sine ratio is positive or negative in that quadrant.In this case, since 60? is the reference angle, students should refer to the 30-60-90 triangle and use the sine ratio for 60?.Therefore the sin 240? is equal to .b) cos 150?c) tan 315?Given a trigonometric ratio, without the use of technology, find all possible values of θ from 0? to 360?.a) Suggested steps:Using the cast rule, draw all possible reference triangles in the coordinate plane.Fill in the ratio (in this case the opposite and adjacent sides) to determine the special triangle & reference angle that will be used to solve the problem.Using the diagram, fill in the reference angle and calculate both of the rotation angles. Check your answer using your calculator.b) c) 300xxDraw a right triangle that includes a 30? angle in standard position. Draw dotted lines to indicate 20?, 10?, and 5?. Discuss what is happening to the lengths of the initial and terminal arm and the triangle height as the angle in standard position approaches 0?. Also discuss what is happening to the second acute angle.The initial and terminal arms are approaching the same length; the height is approaching 0.As the first approaches 0?, the second acute angle approaches 90?. The discussion should include that 0? and 90? are complementary angles. Can both exist in the same right triangle?Determine the primary trigonometry ratios for right triangles with a 0? angle in standard position.Use the same starting triangle. Draw dotted lines to indicate 45?, 60?, 70?, 80?, and 85? angles in standard position. Discuss what is happening to the lengths of the initial and terminal arm and the triangle height.The terminal arm and height are approaching the same length; the initial arm is approaching 0?.Determine the primary trigonometry ratios from the right triangle with a 90? angle in standard position.If you would like to emphasize the limit as the 2 acute angles approach 0? and 90?, consider: . Select [Trig Ratios] [Functions]. Going BeyondUse the unit circle instead of or in addition to triangles. Go beyond 360? and/or deal with negative angles.ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 2.2) Supporting Assessment Glossaryangle in standard position - The location of an angle in the plane in which the vertex is at the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotatequadrantal angle – an angle in standard position where the terminal arm is on the x- or y-axis. Examples are 0°, 90°, 180°, 270° and 360°.reference angle – The acute angle formed by the terminal arm of an angle in standard position and the x-axisrotation angles:positive angle - An angle in standard position swept out by a counterclockwise rotation of its terminal armnegative angle - An angle in standard position swept out by a clockwise rotation of its terminal armreference triangle – A right triangle with a reference angle as one of its verticesOtherLesson 5The Sine LawSTAGE 1BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus.ENDURING UNDERSTANDINGS: Students will understand …Angles in a circle can be expressed in a variety of ways. Primary trigonometric ratios and Pythagorean Theorem only work for right triangles, while sine and cosine law will work for all triangles.ESSENTIAL QUESTIONS: What is triangulation?How is a negative angle possible?How many different ways can you estimate the height of a mountain?Why is it easier to find the exact value of some trigonometric ratios?KNOWLEDGE: Students will know …an angle in standard position, given the measure of the anglethe reference angle for an angle in standard positionthe Pythagorean theorem or the distance formula, can be used to calculate the distance from the origin to a point P (x, y) on the terminal arm of an anglethe sign of a given trigonometric ratio for a given angle, without the use of technology, and explainthe exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?some contextual problems can be solved using trigonometric ratiosSKILLS: Students will be able to …sketch a diagram to represent a problem that involves a triangle without a right anglesolve, using primary trigonometric ratios, a triangle that is not a right triangleexplain the steps in a given proof of the sine law or cosine lawsketch a diagram and solve a problem, using the sine lawLesson SummaryTo teach sine law by showing relationships between the angles in a triangle and their opposite sides.Sketch a diagram and solve a problem, using the sine law Lesson PlanHookConsider allowing students to explore a resource that restricts examples to right angle triangles. At the end of the exploration illicit that there are other kinds of triangles (oblique) that we need to be able to calculate angle and side values for.sourceLesson GoalStudents can see the relationship between angles and the length of opposite sides and use the sine law to find unknown sides and angle measures. Students will realize that sine law can be used when a triangle does not have a 90? angle, but an angle, its opposite side, and at least one other angle or side is known.Activate Prior KnowledgeStudents will be using a protractor to determine angle size(s) and a ruler to determine side length(s). They will need to set up a ratio using appropriate sides and angles.LessonDiscuss the relationship between the sine of the angle and the opposite side. Teach the sine law to show how this relationship can be used to find unknown sides and angles. Stress to students that the sine law can only be used if you are given (or can find) the angle, its opposite side, and at least one other angle or side.Go through examples finding:unknown sidesunknown angles. Provide examples to be given where sine law must be used to solve triangles. Problems should include examples where students must sketch a diagram and solve a problem using the sine law. Going BeyondResourcesMath 20-1 (McGraw-Hill Ryerson: sec 2.3, pages 100-113) Supporting The Sine Law applet can be used for a visual. As you change the length of one side, the other sides and angles change accordingly. The ratios are shown and change accordingly as well. exit slip can be used to test their knowledge using two triangles…one finding an unknown side and one finding the unknown angle. Glossaryambiguous case – From the given information the solution for the triangle is not clear: there might be one triangle, two triangles, or no triangle [Math 20-1 (McGraw-Hill Ryerson: page 104)]oblique triangle – A triangle that is not a right triangleopposite side - The side opposite the reference angleopposite angle – The angle opposite a particular sideratio – A comparison of numbers or quantitiessine law – The lengths of the sides are proportional to the sines of the opposite anglesGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.OtherLesson 6The Cosine LawSTAGE 1BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus.ENDURING UNDERSTANDINGS: Students will understand …Angles in a circle can be expressed in a variety of ways. Primary trigonometric ratios and Pythagorean Theorem only work for right triangles, while sine and cosine law will work for all triangles.ESSENTIAL QUESTIONS: What is triangulation?How is a negative angle possible?How many different ways can you estimate the height of a mountain?Why is it easier to find the exact value of some trigonometric ratios?KNOWLEDGE: Students will know …an angle in standard position, given the measure of the anglethe reference angle for an angle in standard positionthe Pythagorean theorem or the distance formula, can be used to calculate the distance from the origin to a point P (x, y) on the terminal arm of an anglethe sign of a given trigonometric ratio for a given angle, without the use of technology, and explainthe exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?some contextual problems can be solved using trigonometric ratiosSKILLS: Students will be able to …sketch a diagram to represent a problem that involves a triangle without a right anglesolve, using primary trigonometric ratios, a triangle that is not a right triangleexplain the steps in a given proof of the sine law or cosine lawsketch a diagram and solve a problem, using the cosine lawLesson SummaryUsing cosine law when given a triangle with 2 side lengths and included angle size or all 3 side lengthsSketch a diagram and solve a problem using the cosine lawExplain the steps in a given proof of the cosine law. Lesson PlanHookSolve oblique triangles that can be solved with the sine law and a few that require the cosine law. Have students recognize that the sine law will not work for all oblique triangles.Lesson GoalStudents will recognize the scenario in which cosine law would be used, and able to use it to solve a triangle.Activate Prior KnowledgeTeachers may want to review some algebraic manipulations.LessonExplain the steps in a given proof of the cosine law.Draw a triangle that has 2 sides and the enclosed angle. Ask students if the sine law could be used to solve the triangle. Students will realize that because there is not a “pair” (angle with its side), the sine law cannot be used.Cosine law is introduced as the only method to solving a non-right angle triangle with this information.Finding sides:Use c2 = a2 + b2 = 2ab cos C, to find the unknown side length. Once this has been found, the sine law or the cosine law may be used to find other unknown values.Examples should be done to practice finding the unknown side, using the cosine law.Finding angles:Use c2 = a2 + b2 = 2ab cos C, to find the unknown angle. Once this has been found, the sine law can be used to continue, or cosine law may be used again.Examples should be done to practice finding the unknown angles, using the cosine law.Provide examples where cosine law must be used to solve triangles. Problems should include examples where students must sketch a diagram and solve a problem using the cosine law. Going BeyondResourcesMath 20-1 (McGraw-Hill Ryerson: sec 2.4, pages 114-125) Supporting The following applet may be used to show the cosine law while changing lengths of sides or angle measurements Glossarycosine law – the relationship between the cosine of an angle and the lengths of the three sides of any triangle: c2 = a2 + b2 = 2ab cos Coblique triangle – A triangle that is not a right triangleOtherLesson 7The Ambiguous CaseSTAGE 1BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus.ENDURING UNDERSTANDINGS: Students will understand …Angles in a circle can be expressed in a variety of ways. Primary trigonometric ratios and Pythagorean Theorem only work for right triangles, while sine and cosine law will work for all triangles.ESSENTIAL QUESTIONS: What is triangulation?How is a negative angle possible?How many different ways can you estimate the height of a mountain?Why is it easier to find the exact value of some trigonometric ratios?KNOWLEDGE: Students will know …the reference angle for an angle in standard positionthe Pythagorean theorem or the distance formula, can be used to calculate the distance from the origin to a point P (x, y) on the terminal arm of an anglethe exact value of the sine, cosine or tangent of a given angle with a reference angle of 30?, 45? or 60?some contextual problems can be solved using trigonometric ratiosSKILLS: Students will be able to …sketch a diagram to represent a problem that involves a triangle without a right anglesolve, using primary trigonometric ratios, a triangle that is not a right triangleexplain the steps in a given proof of the sine law or cosine lawsketch a diagram and solve a problem, using the sine lawdescribe and explain situations in which a problem may have no solution, one solution or two solutionsLesson SummaryDescribe and explain situations in which a problem may have no solution, one solution, or two solutions.Sketch a diagram and solve a problem using the sine law. Lesson Plan - The Ambiguous CaseHookIf you have a computer, use an applet that shows the ambiguous case. John Scammel created two applets using Geogebra.AmbiguousCase.ggbAmbiguousCase2.ggbfiles were added to the EPSB Understanding by Design share siteSimulate what the applet does, using:straws (uncut, 7 cm, 5 cm, 4 cm, 3 cm, 2 cm) and a protractora geometry set, pencil & paper.Sample straws and protractor activity:Make an approximately 37o degree angle with the uncut and 7 cm straws as shown in the diagram.Check how many triangles can be made with the 2 cm, 3 cm and 4 cm straws. The students should notice that the:2 cm straw gives no solution3 cm straw gives one solution4 cm straw gives two possible solutions. Lesson GoalDescribe and explain situations in which a problem may have no solution, one solution, or two solutions.Sketch a diagram and solve a problem using the sine law.Activate Prior KnowledgeReview the sine law.LessonAfter going through the opening activity, students should understand that there are conditions where there is no solution, one solution, or two solutions, given a specific angle and one fixed side.Define Ambiguous Case.aabhYou may want to go over the general case in your text. Discuss that when a = h, there is one solutionh < a < b there are 2 solutions , there is no solution.Example:In △ABC, <A = 120o, a = 20 cm, b = 15 cm determine all possible values for <B to the nearest degree.(Teacher note: there are 2 answers, 41o and 139o)Do a few more examples, one where the entire triangle is solved.Exit Slip:In △ABC, A = 50o, a = 9.5 cm, b = 7.5 cm. Determine all possible values for C.Recommendation: Do another lesson (Lesson 8) with mixed problem-solving, using sine law, cosine law, ambiguous case and primary trigonometry ratios. Going BeyondResourcesMath 20-1 (McGraw-Hill Ryerson: sec 2.3)Ambiguous Case Video: 4file was added to the EPSB Understanding by Design share site Supporting AssessmentQuestions from text. Glossaryambiguous case – From the given information the solution for the triangle is not clear: there might be one triangle, two triangles, or no triangle [Math 20-1 (McGraw-Hill Ryerson: page 104)]As shown, the given conditions (side a, side b, and A) can produce more than one triangle: an obtuse-angled triangle and an acute-angled triangle.” Mathematics Discovery Dictionaryoblique triangle – A triangle that is not a right triangleopposite side - the side opposite the reference angleopposite angle – the angle opposite a particular sideratio - a comparison of numbers or quantities.sine law – The sides are proportional to the sines of the opposite anglesOther ................
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