Download Unit Three Trigonometry 10 Hours Math 521A

UNIT THREE
TRIGONOMETRY
10 HOURS
MATH 521A
Revised March 20, 01
71
SCO: By the end of grade
11 students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Right triangle trigonometry (8.6)
Engage students in a discussion on solving right triangle trigonometry
problems. Once this review has taken place some real world problems
could be given to the student groups. Later they can brainstorm on
situations 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, not
including the overhang, at what angle of elevation are the rafters and
what is the width of this part of the house?
D17 use trig ratios(and
calculators) to solve a
variety of problems
Sine and Cosine laws (8.6)
Challenge students to try solving a problem where the diagram is an
oblique triangle. (Ex. Math Power 11 p.497 #11 or #16)
It may be worthwhile to show the derivations of Sine and Cosine Law.
C51 derive and use the law
of sines and cosines
72
Worthwhile Tasks for Instruction and/or Assessment
Right triangle trigonometry
Pencil/Paper
Create a problem that uses “angle of depression” and requires
a trigonometric solution.
Communication
Explain how to use a clinometer to determine the height of a
tall structure.
Suggested Resources
Right triangle trigonometry
Math Power 11 p.497 #1-9 odd
Applications
Math Power 11 p.498 #43, 44
Journal
If someone tells you that the tan 500 = 1.1918, explain what
that means in relation to the sides in a right triangle.
Pencil/Paper
Find the maximum height of the ball if its angle of elevation
at the top of its flight path is 380 and it reaches this maximum
height 25m away from where the angle of elevation is
measured.
Note to Teachers: % grade on a hill
on a highway is another practical
application of trigonometry
Sine and Cosine laws
Group Project/Presentation
Create a real world problem where either Sine or Cosine Law
must be used. Explain your problem to the class and present
the solution after they have discussed methods for solving it.
Journal
What must be the known quantities in a triangle before Sine
Law can be used? Cosine Law?
Project
Explain how to determine the height of spire on a building
where the spire is not on the outside face of the building and
sine and/or cosine law must be used to solve the problem.
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Sine and Cosine laws
Math Power 11 p.497 # 11-21,16,
# 35-37,48,49
SCO: By the end of grade
11 students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Sine and Cosine laws (cont’d)
In
ªABC construct an altitude from A to BC and label it “h”.
C51 derive and use the law
of sines and cosines
In the
sin C
B15 derive, analyze and
apply trigonometric
procedures for
calculations in oblique
triangles
two right triangles, find the sin B and
Re-arranging and equating these two yields
h = c sin B
h = b sin C
and c sin B = b sin C
dividing 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 corresponding
side is a constant.
Cosine Law derivation
ª
In ACD
b2 = h2 + x2 and
ªABD
c2 = h2 + (a ! x)2
= h2 + a2 !2ax + x2
= h2 + x2 + a2 !2ax
Replace h2 + x2 with b2
2
2
= b + a !2ax
Replace x with b cos C
c2 = a2 + b2 !2ab cos C
A simple way to think of Cosine Law is that it is basically the
Pythagorean Theorem c2 = a2 + b2 with a correction factor !2ab cos C
that takes into account the fact that you are not working with right
triangles all the time.
In
74
Worthwhile Tasks for Instruction and/or Assessment
Sine and Cosine Law (cont’d)
Pencil/Paper
There is a path from the base of the centre mountain to the
summit. If you have a clinometer, how can you determine the
length of the path up the mountain without actually walking
it.
The width of the base of the mountain is 10 km.
Hint to teachers: They must measure the angle of inclination
and find it is 450.
Journal
When 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/Discussion
At what angle must a carpenter make the top cut on the rafters
for Green Gables if the rafters are 5.5m long to the bird’s
mouth and the width of the
hou
se is 8.5 m. The 8.5 m width is
the
left wing of the house.
Pencil/Paper
Assume the sides of the Point Prim lighthouse will come to a
point at the top of the lighthouse. Using a clinometer, we find
that the sides rise at an angle of 82.50. If the base has a
diameter of 7.4m, find the slant height of the lighthouse sides.
Pencil/Paper
A person sails on a bearing of 0600 for 6 km then turns to a
bearing of 1100 and sails for 9 km. At the end of this second
leg of the triangular course, how far must the person sail to
get back to the starting point and on what bearing must she
sail.
Note to Teachers: Bearings are always with respect to due
North and rotated counterclockwise from there.
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Suggested Resources
Sine and Cosine Law (cont’d)
Math Power 11 p.497 #11,13,15,16,
19,21,35-37,
48,49
SCO: By the end of grade
11 students will be expected
to:
E30 apply the principle of
mathematical induction
C42 create and solve
trigonometric equations
Elaborations - Instructional Strategies/Suggestions
Sine and Cosine Law (cont’d)
Invite students to attempt the problems in the Suggested Resources
column. Have the groups induce that sine law can be used when part
of the given information is an angle and the side opposite that angle.
Similarily, students should be able to induce that cosine law can be
used 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.500-501
Ambiguity 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 given
Example:
Find the measures of angle B and angle A (note: the diagram may not
be accurate).
the
so
lutions are:
pA = 99.70
pB = 50.30
pC = 300
pA = 20.30
pB = 129.70
pC = 300
The first quantity to be calculated is angle B. 50.30 and its supplement
129.70 are both acceptable.
Next get the value of the third angle A. If there are not two sets of
acceptable answers, then in the second column of answers above, pB
+ pC will be greater than 1800, leaving no degrees for pA. If you
demonstrate this scenario by having the students do Math Power 11
p.497 #18 they will see this play out.
Note to Teachers: For a sine law problem(SSA), if the side opposite
the given angle is less than the other side given, then there will be 2
possible solutions or no solutions(sin p A > 1 thus not possible).
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Worthwhile Tasks for Instruction and/or Assessment
Ambiguous case of Sine Law
Group Activity
Create a problem where there are two possible sets of
answers. Give the problem to the class as an exercise then
have a discussion on the solutions the groups arrive at.
Journal
Just by looking at the given information, explain how you can
determine if it an ambiguous case (2 solutions or no solutions)
Suggested Resources
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
Pencil/Paper
A person on shore spots a freighter on a bearing of 0200. He
estimates that the ship is 12 km away. A second person is due
East of the first person and she estimates that the ship is 7 km
away. How far apart are the two people?
Note to Teachers: the side opposite the given angle is 7 and is
less than the other given side, thus this can have 0 or 2
solutions. In this example adjust a compass to radius 7, set it
at point B and we see that it will intersect the base at 2 points
so there are two possible triangles (solutions) ªABD and
ªABC that we must use to find the solutions.
Pencil/Paper
In ªABC pC = 400, c = 4 and b = 10. Find the measure of pB.
Note to Teachers: If you tried to construct this triangle you
would find that the arc drawn from B with radius 4 would not
intersect the base. This problem has no solutions.
Algebraically you would get sin B > 1 which again yields no
solutions.
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Problem Solving Strategies
Math Power 11 p.519 #1,4,5,8