Thursday, January 10, 2013
Monday, October 15, 2012
Trignometry
Subject :
Geometry
Standard :
10th English
Title :
Trigonometry
Content/ Syllabus :
1.
2.
3.
Introduction/Concept:
Right
Triangle
Sine, Cosine and Tangent are all
based on a Right-Angled Triangle
Before getting stuck into the
functions, it helps to give a name to each side of a right triangle:
- "Opposite" is opposite to the angle θ
- "Adjacent" is adjacent (next to) to the angle θ
- "Hypotenuse" is the long one
|
Adjacent is always next to the angle
And Opposite is opposite
the angle
|
 |
Sine,
Cosine and Tangent
Sine, Cosine and Tangent are the three main
functions in trigonometry.
They are often shortened to sin,
cos and tan.
To calculate them:
Divide
the length of one side by another side
... but you must know which sides!
... but you must know which sides!
For a triangle with an angle θ,
the functions are calculated this way:
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Sine
Function:
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sin(θ) = Opposite /
Hypotenuse
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|
Cosine
Function:
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cos(θ) = Adjacent /
Hypotenuse
|
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Tangent
Function:
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tan(θ) = Opposite /
Adjacent
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Example:
What is the sine of 35°?
 |
|
Using this triangle (lengths are
only to one decimal place):
sin(35°) = Opposite / Hypotenuse =
2.8 / 4.9 = 0.57...
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 |
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Good calculators have sin, cos and
tan on them, to make it easy for you. Just put in the angle and press the
button.
But you still need to remember
what they mean!
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Examples
Example:
what are the sine, cosine and tangent of 30° ?
The classic 30° triangle has a
hypotenuse of length 2, an opposite side of length 1 and an adjacent side of
√(3):
Now we know the lengths, we can
calculate the functions:
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Sine
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sin(30°)
= 1 / 2 = 0.5
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Cosine
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cos(30°)
= 1.732 / 2 = 0.866...
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Tangent
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tan(30°)
= 1 / 1.732 = 0.577...
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(get your calculator out and check
them!)
Example:
what are the sine, cosine and tangent of 45° ?
The classic 45° triangle has two
sides of 1 and a hypotenuse of √(2):
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Sine
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sin(45°)
= 1 / 1.414 = 0.707...
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Cosine
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cos(45°)
= 1 / 1.414 = 0.707...
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Tangent
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tan(45°)
= 1 / 1 = 1
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Sohcahtoa
Sohca...what? Just an easy
way to remember which side to divide by which! Like this:
|
Soh...
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Sine = Opposite / Hypotenuse
|
|
...cah...
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Cosine = Adjacent / Hypotenuse
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...toa
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Tangent = Opposite / Adjacent
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...
but please remember "sohcahtoa" - it could help in an exam !
Why?
Why are these functions important?
- Because they let you work out angles when you know sides
- And they let you work out sides when you know angles
Example:
Use the sine function to find "d"
We know
* The angle the cable makes with the
seabed is 39°
* The cable's length is 30 m.
And we want to know "d"
(the distance down).
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Start
with:
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sin 39° = opposite/hypotenuse =
d/30
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Swap
Sides:
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d/30 = sin 39°
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Use
a calculator to find sin 39°:
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d/30 = 0.6293…
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Multiply
both sides by 30:
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d = 0.6293… x 30 = 18.88 to
2 decimal places.
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The
depth "d" is 18.88 m
Exercise
Try this paper-based exercise where you can calculate the sine function for all angles
from 0° to 360°, and then graph the result. It will help you to understand
these relatively simple functions.
Less
Common Functions
To complete the picture, there are 3
other functions where you divide one side by another, but they are not so
commonly used.
They are equal to 1 divided by
cos, 1 divided by sin, and 1 divided by tan:
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Secant
Function:
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sec(θ) = Hypotenuse / Adjacent
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(=1/cos)
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Cosecant
Function:
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csc(θ) = Hypotenuse /
Opposite
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(=1/sin)
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Cotangent
Function:
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cot(θ) = Adjacent /
Opposite
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(=1/tan)
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Links:-
1. More information.
1.
2.
3.
2. Videos.
1.
2.
3.
3. Power point presentation.
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