1-1 Using Trigonometry to Find Lengths

1
1-1Using Trigonometry to Find Lengths
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You have been hired to refurbish the Weslyville
Tower(copy the diagram, 10 lines high, the
width of your page.)
In order to bring enough gear, you need to know
the height of the tower
How would you determine the towers height?
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• When it is too difficult to obtain the
  measurements directly, we can operate on a model
  instead.
• A model is a larger or smaller version of the
  original object.

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• A model must have similar proportions as the
  initial object to be useful.
• Trigonometry uses TRIANGLES for models.
• We construct a similar triangle to represent the
  situation being examined.

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Imagine the sun casting a shadow on the ground.
Turn this situation into a right angled triangle
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The length of the shadow can be measured directly
The primary angle can also be measured directly
X
The Height?
Sooo
40O
200 m
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• Make a model!!
• Draw a right angled triangle with a base of 20 cm
  and a primary angle of 40O, then just measure the
  height!

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• We can generate an equation using equivalent
  fractions to determine the actual height!

General Model Real
X cm
Height
17 cm


Base
20 000 cm
20 cm
0.85
20 000 (0.85) X
170 m X
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In the interest of efficiency..

• Drawing triangles every time is too time
  consuming.
• Someone has already done it for us, taken all the
  measurements, and loaded them into your
  calculator
• Examine the following diagram

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As the angle changes, so shall all the sides of
the triangle.
Recall the Trig names for different sides of a
triangle
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Geometry
hypotenuse
height
base
Trigonometry
hypotenuse
opposite

• O

theta
adjacent
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• Trig was first studied by Hipparchus (Greek), in
  140 BC.
• Aryabhata (Hindu) began to study specific ratios.
• For the ratio OPP/HYP, the word Jya was used

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• Brahmagupta, in 628, continued studying the same
  relationship and Jya became Jiba
• Jiba became Jaib which means fold in arabic

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• European Mathmeticians translated jaib into
  latin
• SINUS
• (later compressed to SIN by Edmund gunter in 1624)

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• Given a right triangle, the 2 remaining angles
  must total 90O.

A 10O, then B 80O A 30O, then B 60O
A
A compliments B
C
B
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• The ratio ADJ/HYP compliments the ratio OPP/HYP
  in the similar mathematical way.
• Therefore, ADJ/HYP is called Complimentary
  Sinus
• COSINE

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The 3 Primary Trig Ratios
SINO opp
hyp
COSO adj
hyp
hyp
opp
TANO opp
adj

• O

adj
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soh cah toa
1

• FIND A

A
X 17
COS25O
17 X
17
1
A 17 X cos25O
17m
A 15.4 m
25O
A
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soh cah toa
1

• FIND A

A
X 12
SIN32O
12 X
12
1
A 12 X SIN32O
12 m
A 6.4 m
A
32O
20
soh cah toa
1

• FIND A

A
X 10
TAN63O
10 X
10
1
A 10 X TAN63O
63O
A 19.6 m
10 m
A
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Tan 40O
X
200
200 (Tan40O) X
168 m X
X
40O
200 m
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Remember Equivalent fractions can be inverted

• 2

5

4
10
4
10

2
5
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• Page 8
• 1,2 a,c
• 3-7

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Find the height of the building
1
X 150
150 X
1
(150) TAN 50 H
HYP
OPP
H
ADJ
50O

• 150 m