Download Triangles: Trigonometry

Triangles: Trigonometry
Right Triangles
Trigonometric Ratios
Rules
Right Triangle Basics
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Pythagorean Theorem
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Pythagorean Triples: Trios of Natural
numbers that are related by the
Pythagorean equality
Multiples: Any multiple of a Pythagorean
Triple is also a Pythagorean Triple
Trigonometric Ratios
Inverse Trigonometry
Problem Solving
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Note: for applied triangle problems such
as these, we will use DEGREES to
measure angles
Minutes and seconds are not in the
curriculum
Later in the unit, we will discuss
RADIANS
Activity
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Exercise 9.2: 2, 3, 4
Vocabulary: Angles of Depression and
Elevation
Triangles: Trigonometry
Sine Rule
Cosine Rule
New Material Today
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Trigonometry for triangles that are NOT
right triangles
Sine Rule (Law of Sines)
Cosine Rule (Law of Cosines)
The Sine Rule
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Useful when an angle and
the side opposite that
angle are known
The numerator/
denominator distinction is
not important as long as
you are consistent
sin A sin B sin C


a
b
c
Sine Rule: Ambiguous Case
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AC is 17 cm
BC is 9 cm
Angle A is 29 degrees
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Solve for all missing sides and angles
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Sine Rule: Ambiguous Case
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AC is 17 cm
BC is 9 cm
Angle A is 29 degrees
Given: angle, opposite and adjacent
sides (NOT the sides that form the
angle)
The two values for the unknown angle
are supplementary
Cosine Rule
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Similar to Pythagorean theorem, with an
adjustment for the fact that the triangles are
not always RIGHT
Note: cos(90) is 0, so the Pythagorean
Theorem is a special case
a  b  c  2bc cos A
2
2
2
b  a  c  2ac cos B
2
2
2
c  a  b  2ab cos C
2
2
2
Cosine Rule
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The cosine rule allows us to find an angle if
all sides are known.
You need only to solve for angle A, though a
formula is given in your formula packet.
b c a
cos A 
2bc
2
2
2
1  b  c  a 
A  cos 

2bc


2
2
2
Activity
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9.5.1: 1-20, multiples of 5
9.5.2: 1-20, multiples of 5
9.5.4: 1-20, multiples of 5
Exercise 9.5.6: 2-8, even
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Skills:
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Use the Sine Rule and Cosine rules to solve
triangles.
Identify and solve the ambiguous case.
Use trigonometry in context
Triangles: Trigonometry
Review of Sine and Cosine Rules
Area
3-D Geometry and Trigonometry
Problem(s) of the Day
Today
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Formula Sheet Reminder, HW Check
Review of:
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Right Angle Trigonometry
Sine Rule and Cosine Rule
Area of Triangles
3-D Geometry and Trigonometry (Quiz
on Tuesday)
Area of a Triangle
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Given: two sides and the angle they
form
Use the triangle on the board (and
trigonometric ratios) to determine a general
formula for the area of a triangle
Hint: in the end, you will need to use the
“one half base times height” definition
Area of a Triangle
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If the “height” or altitude is not given
Given: two sides and the angle they
form
1
A  ab sin C
2
Practice with Area
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Exercise: 9.4
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5, 6, 7
Hint on Parallelograms: Diagonals bisect
each other
3-D Trigonometry
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Look for steps that will allow you to
analyze two-dimensional parts of the
three-dimensional figure
Example 9.3, #1
Practice with 3-D Geometry
and Trigonometry
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Example: 1
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Exercise 9.3:
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2, 3, 4, 5
Homework
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9.3: 2, 3, 4, 5
9.4: 4, 9
Triangles: Trigonometry
Review of Mensuration
Problem of the Day
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From a point A, 150m due south of a
tower, the angle of elevation of the top
of the tower is 30 degrees. From a
point B, due east of the tower, the
angle of elevation of the top of the
tower is 40 degrees.
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Draw a diagram (or two) to display this
How far apart are points A and B
More 2-D  3-D Geometry
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Remember your two-dimensional distance
and midpoint formulae:
A( x1 , y1 ), B( x2 , y2 )
d AB  ( x1  x2 )  ( y1  y2 )
2
M AB
 x1  x2 y1  y2 

,

2 
 2
2
More 3-Dimensional Geometry
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Here are the distance and midpoint formulae
for three dimensions:
A( x1 , y1 , z1 ), B( x2 , y2 , z2 )
d AB  ( x1  x2 )  ( y1  y2 )  ( z1  z2 )
2
M AB
2
 x1  x2 y1  y2 z1  z2 

,
,

2
2 
 2
2
Practice with 3-D Coordinate
Geometry
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Find the length and
the midpoint of the
segment defined by
the points C and D.
C (2,10, 5)
D(14, 0, 45)
Quiz Next Class (Tuesday)
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Print out the formula sheet!!!
Pythagorean Theorem, Angle Sums
Right Triangle Trigonometry
Sine Rule
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Recognize the ambiguous case
Cosine Rule
Area of Triangles
Basic 3-D Problems
Homework
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9.5.6
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8, 14, 18
Challenge: 16
There is a quiz next class. Additional
problems may be selected.