71

UNIT THREE

TRIGONOMETRY

10 HOURS

MATH 521A

Revised March 20, 01

72

SCO: By the end of grade11 students will beexpected to:

D17 use trig ratios(and calculators) to solve a variety of problems

C51 derive and use the law of sines and cosines

Elaborations - Instructional Strategies/SuggestionsRight triangle trigonometry (8.6)Engage students in a discussion on solving right triangle trigonometryproblems. Once this review has taken place some real world problemscould be given to the student groups. Later they can brainstorm onsituations in the real world where trigonometry could be used.

Example:If the height of a gable roof is 2.1m and the rafters are 6.9m long, notincluding the overhang, at what angle of elevation are the rafters andwhat is the width of this part of the house?

Sine and Cosine laws (8.6)Challenge students to try solving a problem where the diagram is anoblique triangle. (Ex. Math Power 11 p.497 #11 or #16)

It may be worthwhile to show the derivations of Sine and Cosine Law.

73

Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Right triangle trigonometryPencil/PaperCreate a problem that uses “angle of depression” and requiresa trigonometric solution.

CommunicationExplain how to use a clinometer to determine the height of atall structure.

JournalIf someone tells you that the tan 500 = 1.1918, explain whatthat means in relation to the sides in a right triangle.

Pencil/PaperFind the maximum height of the ball if its angle of elevationat the top of its flight path is 380 and it reaches this maximumheight 25m away from where the angle of elevation ismeasured.

Sine and Cosine lawsGroup Project/PresentationCreate a real world problem where either Sine or Cosine Lawmust be used. Explain your problem to the class and presentthe solution after they have discussed methods for solving it.

JournalWhat must be the known quantities in a triangle before SineLaw can be used? Cosine Law?

ProjectExplain how to determine the height of spire on a buildingwhere the spire is not on the outside face of the building andsine and/or cosine law must be used to solve the problem.

Right triangle trigonometry

Math Power 11 p.497 #1-9 odd

ApplicationsMath Power 11 p.498 #43, 44

Note to Teachers: % grade on a hillon a highway is another practicalapplication of trigonometry

Sine and Cosine lawsMath Power 11 p.497 # 11-21,16,

# 35-37,48,49

74

SCO: By the end of grade11 students will beexpected to:

C51 derive and use the law of sines and cosines

B15 derive, analyze and apply trigonometric procedures for calculations in oblique triangles

Elaborations - Instructional Strategies/SuggestionsSine and Cosine laws (cont’d)

In ªABC construct an altitude from A to BC and label it “h”.

In the two right triangles, find the sin B andsin C

Re-arranging and equating these two yieldsh = c sin B h = b sin C and c sin B = b sin Cdividing both sides by bc gives

this could easily be extended to

Sine Law in words:For any triangle the ratio of the sine of an angle to its correspondingside is a constant.Cosine Law derivation

In ªACD b2 = h2 + x2 and

In ªABD c2 = h2 + (a ! x)2

= h2 + a2 !2ax + x2 = h2 + x2 + a2 !2ax Replace h2 + x2 with b2

= b2 + a2 !2ax Replace x with b cos C c2 = a2 + b2 !2ab cos CA simple way to think of Cosine Law is that it is basically thePythagorean Theorem c2 = a2 + b2 with a correction factor !2ab cos Cthat takes into account the fact that you are not working with righttriangles all the time.

75

Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Sine and Cosine Law (cont’d)Pencil/PaperThere is a path from the base of the centre mountain to thesummit. If you have a clinometer, how can you determine thelength of the path up the mountain without actually walkingit.The width of the base of the mountain is 10 km.

Hint to teachers: They must measure the angle of inclinationand find it is 450.JournalWhen is it to your advantage to use sine law in each form:ratios of; side : sin of an angle or ratios of: sin of an angle : side.

Pencil/Paper/DiscussionAt what angle must a carpenter make the top cut on the raftersfor Green Gables if the rafters are 5.5m long to the bird’smouth and the width of the house is 8.5 m. The 8.5 m width is theleft wing of the house.

Pencil/PaperAssume the sides of the Point Prim lighthouse will come to apoint at the top of the lighthouse. Using a clinometer, we findthat the sides rise at an angle of 82.50. If the base has adiameter of 7.4m, find the slant height of the lighthouse sides.

Pencil/PaperA person sails on a bearing of 0600 for 6 km then turns to abearing of 1100 and sails for 9 km. At the end of this secondleg of the triangular course, how far must the person sail toget back to the starting point and on what bearing must shesail.

Note to Teachers: Bearings are always with respect to dueNorth and rotated counterclockwise from there.

Sine and Cosine Law (cont’d)

Math Power 11 p.497 #11,13,15,16, 19,21,35-37, 48,49

76

SCO: By the end of grade11 students will be expectedto:

E30 apply the principle of mathematical induction

C42 create and solve trigonometric equations

Elaborations - Instructional Strategies/SuggestionsSine and Cosine Law (cont’d)Invite students to attempt the problems in the Suggested Resourcescolumn. Have the groups induce that sine law can be used when partof the given information is an angle and the side opposite that angle.Similarily, students should be able to induce that cosine law can beused when given: < 2 sides and the included angle (SAS) < all 3 sides (SSS)Ambiguous case of Sine Law (8.7)Invite students to do the Investigation on p.500-501Ambiguity creeps into play when you are given: < one acute angle < the side opposite this angle and it is smaller than the second side that is givenExample:Find the measures of angle B and angle A (note: the diagram may notbe accurate).

the solutions are:

pA = 99.70 pA = 20.30

pB = 50.30 pB = 129.70 pC = 300 pC = 300

The first quantity to be calculated is angle B. 50.30 and its supplement129.70 are both acceptable.Next get the value of the third angle A. If there are not two sets ofacceptable answers, then in the second column of answers above, pB+ pC will be greater than 1800, leaving no degrees for pA. If youdemonstrate this scenario by having the students do Math Power 11p.497 #18 they will see this play out. Note to Teachers: For a sine law problem(SSA), if the side oppositethe given angle is less than the other side given, then there will be 2possible solutions or no solutions(sin p A > 1 thus not possible).

77

Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Ambiguous case of Sine LawGroup ActivityCreate a problem where there are two possible sets ofanswers. Give the problem to the class as an exercise thenhave a discussion on the solutions the groups arrive at.

JournalJust by looking at the given information, explain how you candetermine if it an ambiguous case (2 solutions or no solutions)

Pencil/PaperA person on shore spots a freighter on a bearing of 0200. Heestimates that the ship is 12 km away. A second person is dueEast of the first person and she estimates that the ship is 7 kmaway. How far apart are the two people?

Note to Teachers: the side opposite the given angle is 7 and isless than the other given side, thus this can have 0 or 2solutions. In this example adjust a compass to radius 7, set itat point B and we see that it will intersect the base at 2 pointsso there are two possible triangles (solutions) ªABD andªABC that we must use to find the solutions.

Pencil/PaperIn ªABC pC = 400, c = 4 and b = 10. Find the measure of pB.

Note to Teachers: If you tried to construct this triangle youwould find that the arc drawn from B with radius 4 would notintersect the base. This problem has no solutions.Algebraically you would get sin B > 1 which again yields nosolutions.

Ambiguous case of Sine Law

Math Power 11 p.510 #8,10,12,17, 20,22,28,29

Applications

Math Power 11 p.511 #31

Problem Solving StrategiesMath Power 11 p.519 #1,4,5,8