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 sideview 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 sideview?
OPPOSITE SIDE/ HYPOTENUSE
= FRONT VIEW/ SIDEVIEW.
~~ Seeing this way can help you
understand why the formula.
~~ The frontview is always
influenced by the inclination of the object. So apparently when an object tilts,
its front view reduces.
~~ The sideview is the actual
length of the stick. The sideview 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.
SIN
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 yaxis, you plot the speed of the car.
On the xaxis 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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