Sunday, 9 June 2013

TRIGONOMETRY: The True Purpose of Sin,Cos,Tan and why we

Visualizing Trignometry.

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  0 an object is 0 % vertical. Hence sin(0)=0.
o   At 90 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 90…full shadow.                 At 30……half the shadow.      At 0…………no shadow.

                                  Analogy 2:

Imagine a missile  is thrown on ground. If the missile falls at 90….the whole energy of the missile will be passed to the ground.   
        Collision at 90                Collision at 60             Collision at 30                  At 0
§  Let’s say the missile falls with a force of 400KN. At 90 all its energy goes to ground.
§  However what if the missile hits at 30? 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  30  Force (or pressure ) is half as compared to 90.
§  Hence sin(30)=1/2.

§  Note:
§  At 0  , The missile moves parallel to the ground and hence impact is 0.Hence sin(0)=0.

§  Cos is a measure of how much horizontal (or parallel to the surface under consideration) an object is.
§  Hence cos(0)=1 (meaning at 0 the object is fully horizontal)
§  Hence cos(90)=0 (meaning at 90 the object is fully vertical and hence 0% horizontal).
§  Cos (60)=1/2 meaning at 60 an object has ½ as influence as when it was fully horizontal (i.e at 0).
§  Cos(180)=-1 meaning that the object is fully horizontal but in the opposite direction as it was at 0.
                           Analogy 1

A Stick held below a light bulb will……….
        cossine.png                cos.jpg                            
put maximum shadow on the ground             At 90 the shadow will be absent on the ground.
when at 0.

                        Alternate way to look at Trignometry
Angle (Ɵ)
Percentage Form

           1 / 2
1 / 2  * 100  =  50%
50%  Vertical
            1 /
1 /  =  70%
70%  Vertical
              / 2
    / 2 * 100  =  86%
86%  Vertical
    1 * 100  = 100%
100%   Vertical
Sin (180)
     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 0 its slope is 0%.
Hence Tan (0)=0.
Now imagine a slide at 90. 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)=

Negative values of Tan(Ɵ) means that the angle is more than 90. (-ve sign plays the role of  and indicator indicating to you that the slope is beyond 90.

                         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.
§  If  Speed was plotted on the Y-axis of the above graph, then ,Tan(Ɵ) would be Acceleration of the car.
§  This happens because,
§  Acceleration is rate of change of speed.
§  Just as Speed is rate of change of Distance.
§  Tan(Ɵ)  itself in general is a measure of rate of change of Y-axis quantity w.r.t  X-axis quantity.

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.
Car Downslope.png
§  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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