1.This page assumes that you understand basic concepts like functions, Cosine, Sine, Pi, Degree and Radians, Triangle, Pythagoras theorem, Cartesian axis. This is a brief recap for those who forgot.
This is a Cartesian rectangular axes.
Horizontal line is x axis.
Vertical line is y axis.
Pi = Circumference / Radius of circle
One Radian = Circumference / Radius of circle= 1
Circumference = Radius of circle= Unit circle
Cos x = Adjacent side / Hypotenuse of Triangle
Sin x = Opposite side / Hypotenuse of Triangle
Tan x = Opposite side / Adjacent side
2.Conversion between degree and Radian
Pi / 180 X Degree = Radian angle
180/ Pi X Radian = Degree angle
Degree and Radian will be used interchangeably.
Graphing Cosine function
Let us begin by examining cosine function,
[ Why? because it is easy, If I introduce asymptote of Tan you may get so frightened,that you may stop reading this page...]
Let us begin by looking at growth of Cos function.
y = f(x ) = Cos x
Mind you x has to be in Radians, y will be cos x, so we get two points to plot on graph paper.
Why Radians?
Which is easier to plot 90, 180, 360, 720, 270 on bit of paper or 3.14, 1.7, 6.28? Does 270, 360,180 degree show us any relationship to circle?
Radian is much closer to real life circumstances
Circular Relationship is what we are trying to understand.
But computing Pi/ 2 or Pi / 3 is still a pain in the a++!
Now Cos 0 = 1
Imagine a Flat line lying on X axis, so adjacent side = Opposite side
Cos 90 = 0
Imagine a flag pole all it is made of is a hypotenuse, no side, so zero / Hypotenuse = 0
So cos decreases from 1 to 0 in Pi / 2 angle
[or 90 degree angle]
Why Do you need to believe me?
Value of Cos 30 degree = Sq rt 3 / 2 =0.866
Cos 45 degree = 1 / Sq rt 2 = 0.707
Cos 60 degree = 1 / 2 = 0.5
See the cos value is falling down from 1,0.8 to 0.5; 0.
The plot of this on graph will be like this.
Understanding Cos Function from 90 to 180 degree
At 90 cos is 0.
What happens when cos goes beyond 90?
Look at the graph below, the adjacent side which is X axis is negative, Hypotenuse = sq rt ( square of adjacent side + square of Opposite side) This HAS be positive.
Why?
Boss get your algebra right...
[Negative adjacent side square ]= [ Minus] 2 = Plus square root of which must be positive.
Unless you want COMPLEX numbers which really complicates the simple matter....
So believe me when I say
Cos x = Negative / Positive = Negative for x 90 to 180 degree (90,180 )
So what is Cos 180 degree?
Flat line prostate on Negative X axis Adjacent = Hypotenuse both are same with Adjacent side negative so we conclude
Cos 180 = - 1
So cos decreases further from 0 to - 1 in 90 to 180 degrees.
So the theorem.
In first two quadrants=Cos always falls.
Let us plot Cos now
Cos 120 =
Cos 90 + 30 = Cos ( 180 - 30 ) = Signed Cos 60 = - 1 / 2 = - 0.5
Cos 135 = Cos ( 180 - 45) = Signed Cos 45 = - 0.707
Cos 150 = Cos ( 180 - 60 ) = Signed Cos 30 = - 0.866
So cos falls from 0 towards - 1 in next 90 degrees, 90 to 180.
Setting up the graph in Radians we get the picture, which resembles a wave like function....
Note that the X axis is in radians, y axis is value of cos.
Let us plot Cos 180 to Cos 270 degrees.
Cos 180 is - 1.
What is Cos 270?
Look at negative x axis, negative y axis, Yes; Cos 270 is Negative, Cos is Negative in 3rd quadrant.
So 270 degree angle looks like a mirror image of 90 degree as shown [ shadow of a flag pole]. All hypotenuse, zero side triangle = - 0
Cos 270 = - 0
Let us find some plot points
Cos 210 = Cos 180 + 30 = Signed Cos 30 = - Sq rt 3 / 2 = - 0.866
Cos 225 = Cos 180 + 45 = Signed Cos 45 = - 0.707
Cos 240 = Cos 180 + 60 = Signed Cos 60 = - 0.5
Observe Cos is now increasing from - 1 towards 0. In 3 rd quadrant Cos increases.
Plotting on graph we get.
Note x axis is in radian angles, as it is convenient to use Rads to quantify angles.
4th Quadrant
So on for 4rd quadrant, but observe adjacent side becomes positive again in 4 quadrant, as it now, again lies in positive x axis. Cos x , yes is positive again.
Cos 360 = Cos 0 = 1 all adjacent side = same as hypotenuse.
Cos 300 = Cos 360 - 60 = Signed Cos 60 = 0.5
Cos 315 = Cos 360 -45 = Signed cos 45 = 0.707
Cos 330 = Cos 360 - 30 = Signed Cos 30 = 0.866
You know cos is headed towards 1, thus completing the cycle.
RULE
1St and 2 Quadrants=Cos falls from 1 to 0 to - 1.
3rd and 4th Qudrants=Cos raises from - 1 to 0 to 1.
Now what about 390 degree, simply 360 + 30 is Cos 30 only. This can be repeated ad nauseam till infinity, thus creating a Cosine wave that stretches from Negative infinity to Positive infinity, touching 1 at zero rads. Finally the total area covered by Cosine wave from 0 degree to 360 degree [ 2Pi rads] is a full circle, wrapped around the x axis like shown in the figure.
Next
Manipulation| Modulation of Cosine wave.
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Email me at minerva.santh@gmail.com
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