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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 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.
The length of the shadow can be measured directly
The primary angle can also be measured
directly
The Height?
200 m 40O
X
Sooo…
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!
We can generate an equation using equivalent fractions to determine the actual height!
HeightBase
= 17 cm20 cm
= 20 000 cm
X cmGeneral Model Real
0.85 = 20 000 cm
X cm
20 000 (0.85) = X170 m = X
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
O O OO
As the angle changes, so
shall all the sides
of the triangle.
Recall the Trig names for different sides of a triangle…
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
BC
A “compliments” B
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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