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1-1 Using Trigonometry to Find Lengths

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1-1 Using Trigonometry to Find Lengths

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1-1Using Trigonometry to Find Lengths

In order to bring enough gear, you need to know the height of the tower……

How would you determine the tower’s height?

- 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.

- 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.

Turn this situation into a right angled triangle

The primary angle can also be measured directly

X

The Height?

Sooo…

40O

200 m

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!

X cm

=

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

- 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

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…

hypotenuse

height

O

base

Trigonometry

hypotenuse

opposite

“theta”

adjacent

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

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:

SINUS

(later compressed to SIN by Edmund gunter in 1624)

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

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

Therefore, ADJ/HYP is called “Complimentary Sinus”

COSINE

SINO = opp

O

hyp

COSO = adj

hyp

hyp

opp

TANO = opp

adj

adj

1

A

X 17

FIND A:

17 X

COS25O =

17

1

A = 17 X cos25O

17m

A = 15.4 m

25O

A

1

A

X 12

FIND A:

12 X

SIN32O =

12

1

A = 12 X SIN32O

12 m

A = 6.4 m

A

32O

1

A

X 10

FIND A:

10 X

TAN63O =

10

1

A = 10 X TAN63O

63O

A = 19.6 m

10 m

A

X

200

200 (Tan40O) = X

168 m = X

X

40O

200 m

2

5

=

4

10

4

10

=

2

5

Page 8

[1,2] a,c

3-7

TAN 50 = H

150

1

150 X

X 150

Find the height of the building

1

(150) TAN 50 = H

HYP

OPP

H

150 m

ADJ

50O