# Cosine Graph

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**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.

**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

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Graph Dictionary

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