Visualizing Trignometry.
SIN
o
Sin
is a measure of how much vertical (or perpendicular to the base surface) an object(OR
graph’s slope OR force) is.
o At 0⁰
an object is 0 % vertical. Hence sin(0)=0.
o
At
90⁰ an object is 100% vertical.
Hence sin(90)=100%=100/100=1.
 Analogy 1.
Imagine a light bulb kept at
your one hand and lets say there is a
stick in your other hand.
Now look at the shadow of that
stick on the wall.
The more vertical ( or
Perpendicular to the surface on which it is standing) the stick is the more
the shadow.
  
At 90⁰…full shadow. At 30⁰……half the shadow. At 0⁰…………no
shadow.
|
Analogy 2:
Imagine a missile is thrown on ground. If the missile falls at
90⁰….the whole energy of the missile
will be passed to the ground.
Collision at 90⁰
Collision at 60⁰ Collision at 30⁰
At 0⁰
§
Let’s
say the missile falls with a force of 400KN. At 90⁰ all its energy goes to ground.
§
However
what if the missile hits at 30⁰? Will all the 400KN force be
transferred to the ground or only a fraction(a small part) of it?
§
This
fraction(part) is sin(30)=1/2. Thus now the force transferred to the ground
(impact on the ground) is ½ *400KN= 200KN.Conclusion
§
At 30⁰ Force (or pressure ) is half as compared to
90⁰.
§
Hence
sin(30)=1/2.
§ Note:
§ At 0⁰ , The
missile moves parallel to the ground and hence impact is 0.Hence sin(0)=0.
COS
§
Cos
is a measure of how much horizontal (or parallel to the surface under
consideration) an object is.
§ Hence cos(0)=1 (meaning at 0⁰ the object is fully horizontal)
§ Hence cos(90)=0 (meaning at 90⁰ the object is fully vertical and hence 0%
horizontal).
§ Cos (60)=1/2 meaning at 60⁰ an object has ½ as influence as when it was fully
horizontal (i.e at 0⁰).
§
Cos(180)=-1
meaning that the object is fully horizontal but in the opposite
direction as it was at 0⁰.
Analogy 1
A
Stick held below a light bulb will……….
 
put
maximum shadow on the ground
At 90⁰ the shadow will be absent on
the ground.
when
at 0⁰.
|
Alternate way to look
at Trignometry|
Angle (Ɵ)
|
Value
|
Percentage
Form
|
Meaning
|
|
Sin(30)
|
 1 / 2 |
1 / 2
* 100 = 50%
|
50%
Vertical
|
|
Sin(45)
|
1 /
 |
1 /
 = 70% |
70%
Vertical
|
|
Sin(60)
|
 / 2 |
/ 2 * 100 =
86% |
86%
Vertical
|
|
Sin(90)
|
1
|
1 * 100 = 100%
|
100%
Vertical
|
|
Sin (180)
|
0
|
0 * 100 = 0%
|
0%
Vertical
|
Similarly
for Cos a table can be made.
What about TAN?
TAN is a
measure of how much sloping an
object (or graph) is.
|
Analogy 1
Imagine
a slide. At 0⁰ its slope is 0%.
Hence
Tan (0)=0.
Now
imagine a slide at 90⁰. Can you still call it a
slide? What will happen to a person who comes down such a slope?
In
mathematical terms we say that its
slope is infinite. Hence Tan(90)=
Note:
Negative
values of Tan(Ɵ) means that the angle is more than 90⁰. (-ve sign plays the role
of and indicator indicating to you
that the slope is beyond 90⁰.
|
Analogy 2
[Graphical meaning of tan(Ɵ)]
Imagine
that on the y-axis, you plot the speed of the car.
On
the x-axis you plot the time taken by the car to reach that speed.
Now ,
TAN(Ɵ)=acceleration of the car.
SIN(Ɵ)=Final
speed of the car.
COS(Ɵ)=
Total Time taken by the car.
|
And SEC , COT AND COSEC?
Well
in some cases….some things vary inversely with respect to sin(Ɵ), cos(Ɵ) and
tan(Ɵ).
 Analogy 1
For example the more sloping a
road (downhill) is……the less the
friction between the road and a rock rolling along the road.
§
Friction
 1/tan(Ɵ)......[1]
The above equation [1] can also be written as follows
§
Friction
cot(Ɵ)…………[2][where cot(Ɵ)=1/tan(Ɵ)]. |
Final Notes:
§
The
max value of sin,cos,sec or cosec is ‘1’.
§ ‘1’ indicates full or complete
(Full shadow, Full force etc.)
§ Minimum is ‘0’(absence of that
component).
§ All other values hang between ‘0’
and ‘1’.
§ Negative values indicate angles
greater than 90 Degrees.
§ All the value of sin,cos and tan
are ratios.
§ These ratios are functions (i.e
they keep changing) with the values of Ɵ.
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