Unit 7 notes docx - St John Brebeuf

Apprenticeship and workplace Math 10
St.John Brebeuf
Unit 7:
Trigonometry of Right
Triangles.
Mathematics
Department
Goals:
In this unit you will be applying prior knowledge about triangles
and similar figures to:
o Determine the trigonometric ratios
o Use these ratios to determine the lengths of sides
o Use these ratios to determine the measure of angles
Key Terms:
Angle of depression
Angle of elevation
Cosine
Hypotenuse
Leg
Pythagorean Theorem
Sine
Tangent
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Apprenticeship and workplace Math 10
St.John Brebeuf
Unit 7.1:The Pythagorean Theorem.
‣ Right Triangle: A triangle with one right angle
‣ Hypotenuse: The longest side of a right triangle opposite the 90
angle.
‣ Pythagorean Theorem: In a right triangle, the sum of the
squares of the lengths of the arms is equal to the square of the
hypotenuse
‣
Hypotenuse
arm
arm
a
c
b
a 2 + b 2 = c 2
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Apprenticeship and workplace Math 10
St.John Brebeuf
Example 1. Calculate the length of BC
25f
t
A
C
30ft
B
Solution:
Example 2.
3.8m x
a) Calculate the length of x
2.5m
b) Calculate the length of y
Y 6.8m
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Apprenticeship and workplace Math 10
St.John Brebeuf
Solution:
a)
b)
‣ Pythagorean Triple: any set of 3 numbers that satisfy the
Pythagorean Theorem.
Some common sets are:
3,4,5 since 3 2 + 4 2 = 5 2
5,12,13 since 5 2 + 12 2 = 13 2
12,16,20 since 12 2 + 16 2 = 20 2
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Apprenticeship and workplace Math 10
St.John Brebeuf
Example 3.
Marc is going to paint the exterior of his house. He has a 40- foot
ladder and knows that for safety reasons the base of the ladder must
be between 9 and 12 feet from the base of the wall.
What are the maximum and minimum heights the ladder will reach up
the wall
Solution:
Maximum Height:-
Minimum Height:-
40ft
40ft
9ft
12ft
Complete notebook assignment page 278 # 1-7
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Apprenticeship and workplace Math 10
St.John Brebeuf
Unit 7.2: The Sine Ratio.
Opposite
Hypotenuse
A
Adjacent
Sin A =
SOH
Put your calculator into degree mode.
Find:
a) sin 20 b) sin 55 c) sin 84 d) sin 5
= = = =
Example 1. Finding a Side
3.2m r
23
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Apprenticeship and workplace Math 10
St.John Brebeuf
Calculate the length of r in the diagram above
Solution:
3.2m r
23
‣ Angle of Depression: The angle formed between the horizontal
and the line of sight looking downwards.
Angle of depression
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Apprenticeship and workplace Math 10
St.John Brebeuf
‣ Angle of Elevation: The angle formed between the horizontal and
the line of sight looking upwards, sometimes referred to as the
angle of inclination.
Angle of elevation
Example 2.
From the top of a cliff by the ocean, Cedric sights a boat at an angle
of depression of 48°. If the top of the cliff is 73m above the surface
of the water and Cedric is 2m tall, how far is Cedric from the boat
48 °
Cliff
Cedric
48 °
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Apprenticeship and workplace Math 10
St.John Brebeuf
Solution:
Complete notebook assignment page 289 # 1-8
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Apprenticeship and workplace Math 10
St.John Brebeuf
Unit 7.3: The Cosine Ratio.
Opposite
Hypotenuse
A
Adjacent
Cos A =
CAH
Put your calculator into degree mode.
Find:
a) cos 20 b) cos 55 c) cos 84 d) cos 5
= = = =
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Apprenticeship and workplace Math 10
St.John Brebeuf
Example 1.
Calculate the length of q
10°
q
6.1m
Solution:
Example 2. Calculate the length of p and r in the triangle below.
4.3cm
r
51°
p
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Apprenticeship and workplace Math 10
St.John Brebeuf
Solution:
Example 3.
In construction Marie knows that a force acting at an angle can be
broken up into a vertical force and a horizontal force. If a force of
365 Newtons is exerted diagonally downwards at an angle of 30° to the
horizontal what force will be applied horizontally
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Apprenticeship and workplace Math 10
St.John Brebeuf
Solution:
365N
30°
x
Complete notebook assignment page 297 # 1-6
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Apprenticeship and workplace Math 10
St.John Brebeuf
Unit 7.4: The Tangent Ratio.
Opposite
Hypotenuse
A
Adjacent
Tan A =
TOA
Put your calculator into degree mode.
Find:
a) tan 20 b) tan 55 c) tan 84 d) tan 5
= = = =
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Apprenticeship and workplace Math 10
St.John Brebeuf
Example 1.
Gull Harbour Lighthouse is located on Lake Winnipeg, Manitoba.
The lighthouse is 14.6m tall and stands
7m above the surface of the lake.
The angle of depression to a boat on
Lake Winnipeg is 27.
How far away is the boat from the base of the lighthouse
Solution:
27
14.6m
h
27
7m
x
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Apprenticeship and workplace Math 10
St.John Brebeuf
Example 2.
A dockworker pulls a light crate (measuring 2m x 2m x 2m) up to the
dock using a pulley system. The angle of elevation of the rope is 50.
The man is 2m from the edge of the pier and the bundle clears the pier
by 0.5m.
Hoe close are the pulleys to each other when the bottom pulley is at
hand level
Solution:
Draw a simple diagram
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Apprenticeship and workplace Math 10
St.John Brebeuf
h
50
d
2m
2m
2m
0.5m
Complete notebook assignment page 305 # 1-7
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Apprenticeship and workplace Math 10
St.John Brebeuf
Unit 7.5: Finding Angles and Solving Right Triangles.
Example 1.
Determine the angle indicated in each of the following.
a) A guy wire 8.5m long is attached 5.7m from the base of a pole.
b) The angle of depression from a point 10.1m down a hill if the
horizontal distance is 6.9m
c) The angle between the side of a house and the glass roof of a small
bay window, if the bay window is 75 inches deep and the vertical
displacement of the roof is 42 inches.
Solution:
a) C
8.5m
B
A
5.7m
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Apprenticeship and workplace Math 10
St.John Brebeuf
6.9m
b) X Y
10.1m
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Apprenticeship and workplace Math 10
St.John Brebeuf
c) A
42in
B 75in C
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Apprenticeship and workplace Math 10
St.John Brebeuf
Example 2.
Mike needs to replace a set of steps at the front of his house.
He knows that the distance between the ground and the first floor
landing is 0.86m and that the stairs end at a point 1.2m from the edge
of the landing.
a) What will be the angle of elevation from the bottom to the landing
b) What is the distance between the bottom of the stairs and the
landing
Solution:
a)
landing
0.86m Ground
1.2m
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Apprenticeship and workplace Math 10
St.John Brebeuf
b) Use Pythagorean Theorem
Complete Notebook Assignment page 311 # 1-8
Complete Unit 7 Review page 316 # 1-9
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Apprenticeship and workplace Math 10
St.John Brebeuf
Reflect on your learning
Now that you have completed this unit check the box that applies
to you
RED AMBER GREEN
I understand all the key terms.
I can use Pythagorean Theorem
to find the missing sides in right
triangles
I can determine which
trigonometric rations to use
in a given situation
I can use trigonometric ratios
to find missing sides and angles
I can solve problems in context
using trigonometry.
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Apprenticeship and workplace Math 10
St.John Brebeuf
I have completed all
homework assignments.
I have attended lunchtime
tutorials for extra help.
I am ready to sit my
unit 7 test.
Target:
In my Unit Test I hope to achieve
%
Student’s Signature ____________________
Date__________
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