TRIGONOMETRY: The True Purpose of Sin,Cos,Tan and why we use and study them.

WHY IS SIN (ANGLE) = OPPOSITE SIDE/ HYPOTENUSE.

Example 1

|~~||  Imagine a stick kept between a bulb and a wall.

The front view of the stick

|~~||  Now the shadow that falls of the stick upon the wall is like the front view of the wall. Now lets say the stick is inclined at 30 degree,  only a small shadow of the stick falls upon the wall.

|~~|| In other words,  the wall presents  the opposite side view of the slanting  stick and thus gives us a measure of how much the length of the stick appears to be reduced due to the inclination.

The side-view of the stick

|~~|| Now if you look at the stick from the side, you are obviously going to see the actual length of the stick, aren’t  you.

|~~|| No  matter how much the stick is inclined,  does  it  change the side-view?

OPPOSITE SIDE/ HYPOTENUSE = FRONT VIEW/ SIDE-VIEW.

|~~|| Seeing this way can help you understand why the formula.

|~~|| The front-view is always influenced by the inclination of the object. So apparently when an object tilts, its front view reduces.

|~~|| The side-view is the actual length of the stick. The side-view is completely unaffected by the slope of the stick.

Thus the formula can be written as 

APPARENT LENGTH OF THE STICK AS SEEN FROM FRONT / ACTUAL LENGTH OF THE STICK.

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. Note §  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.

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.