# Cosine Graph

We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.

Affiliations with Schools & Educational institutions are also welcome.

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

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

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.

Example :
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

Graph Dictionary