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1-1 Using Trigonometry to Find Lengths. 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 tower’s height?.

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1 1 using trigonometry to find lengths

1-1Using Trigonometry to Find Lengths


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 tower’s height?




Imagine the sun casting a shadow on the ground
Imagine the sun casting a shadow on the ground. to be useful.

Turn this situation into a right angled triangle


The length of the shadow can be measured directly
The length of the shadow can be measured directly to be useful.

The primary angle can also be measured directly

X

The Height?

Sooo…

40O

200 m


Make a model!! to be useful.

Draw a right angled triangle with a base of 20 cm and a primary angle of 40O, then just measure the height!


X cm to be useful.

=

20 000 cm

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


In the interest of efficiency
In the interest of efficiency.. to be useful.

  • 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


O to be useful.

O

O

O

As the angle changes, so

shall all the sides

of the triangle.

Recall the Trig names for different sides of a triangle…


Geometry
Geometry to be useful.

hypotenuse

height

O

base

Trigonometry

hypotenuse

opposite

“theta”

adjacent


Trig was first studied by Hipparchus (Greek), in 140 BC. to be useful.

Aryabhata (Hindu) began to study specific ratios.

For the ratio OPP/HYP, the word “Jya” was used


Brahmagupta, in 628, continued studying the same relationship and“Jya” became “Jiba”

“Jiba became Jaib”which means “fold” in arabic


European Mathmeticians translated “jaib” into latin: relationship and

SINUS

(later compressed to SIN by Edmund gunter in 1624)


Given a right triangle, the 2 remaining angles must total 90 relationship andO.

A = 10O, then B = 80O

A = 30O, then B = 60O

A

A “compliments” B

C

B


The ratio ADJ/HYP compliments the ratio OPP/HYP in the similar mathematical way.

Therefore, ADJ/HYP is called “Complimentary Sinus”

COSINE


The 3 primary trig ratios
The 3 Primary Trig Ratios similar mathematical way.

SINO = opp

O

hyp

COSO = adj

hyp

hyp

opp

TANO = opp

adj

adj


Soh cah toa
soh cah toa similar mathematical way.

1

A

X 17

FIND A:

17 X

COS25O =

17

1

A = 17 X cos25O

17m

A = 15.4 m

25O

A


Soh cah toa1
soh cah toa similar mathematical way.

1

A

X 12

FIND A:

12 X

SIN32O =

12

1

A = 12 X SIN32O

12 m

A = 6.4 m

A

32O


Soh cah toa2
soh cah toa similar mathematical way.

1

A

X 10

FIND A:

10 X

TAN63O =

10

1

A = 10 X TAN63O

63O

A = 19.6 m

10 m

A


Tan 40 o

X similar mathematical way.

Tan 40O =

200

200 (Tan40O) = X

168 m = X

X

40O

200 m


Remember equivalent fractions can be inverted
Remember: Equivalent fractions can be inverted similar mathematical way.

2

5

=

4

10

4

10

=

2

5


Page 8 similar mathematical way.

[1,2] a,c

3-7


Find the height of the building

TAN 50 = H similar mathematical way.

150

1

150 X

X 150

Find the height of the building

1

(150) TAN 50 = H

HYP

OPP

H

150 m

ADJ

50O


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