Cosine Graph : Trigonometric function `cos x` is a periodic functions. Cosine function has maximum value as +1 and minimum as -1.
Since cos(-x) = cos x is an even function.
The value of the functions `cos x` repeats in every 2π units of `x` .
Therefore this function is periodic with a period of 2π
Some definitions used in cosine curve are Amplitude: This is half the distance between the maximum and minimum values.
The term `a` in the general form of the sinusoidal function represents the amplitude of this function.
The amplitude is a constant for sinusoidal functions. Period: This is the smallest time needed for a function to execute one complete cycle.
From the general form of cosine function, the period T can be written as T = 2π/ b
Now let us come back to the simplest cosine function .
y =a cos bx
where a = amplitude .
Period = 2π / b Cos (0) = 1 so cosine graph always start from 1.
As b gets larger, the period decreases.
Phase shift of cosine function
y = a cos (bx + c ) + d
Both b and c affect the phase shift or movement of graph
If c -----------> Positive -----------> the shift is towards right
If c -----------> Negative -----------> shift is towards left.
If d ----------->Positive ----------->graph shifted up by d units.
If d ----------->Negative ---------> graph shifted down by d units.
Phase shift = - c / b
Period = 2π / b
The phase shift is the amount that the curve is moved in a horizontal direction from its normal position.
Phase shift, Negative -----------> Left Displacement
Phase shift, Positive -----------> Right Displacement
To find the phase shift , just make
bx + c = 0 and solve.
y = 2 cos (2x +1 )
Here amplitude = a = 2
2x + 1 = 0 ⇒ x = -1/2 so Phase shift = -1/2.
Period = 2π / b
Period = 2π / b = 2π / 2 = π cosine graph