The frequency factor of b = 1 is of Option "A","C","D" and "G"
Angular frequency is the rate at which an object moves through some number of radians. If you know the time it took for the object to move through an angle, the angular frequency is the angle in radians divided by the time it took.
Formula for angular frequency
ω = 2 π / T
where
ω = angular frequency
T = Time period
According to the question
The following have a frequency factor of b = 1
i.e
b = Angular frequency = 1
By using Angular frequency formula
ω = 2 π / T
1 = 2 π / T
T = 2 π
Therefore , when T = 2 π then only b=1
Option A: Sine function whose period is 2π radian
This statement is true .
The period of the sine function is 2π, which means the value of the function is the same every 2π units .
i.e
T = 2π
Therefore it will have frequency factor of b = 1
Option B: A sine function whose x coefficient is 2π
This statement is False .
Option C: A cosine function with no phase shift whose x-coefficient is 1.
This statement is true .
As per definition of phase shift :
The phase shift describes how far horizontally the graph has been moved from the regular sine or cosine. As such, the value is equal to 0 i.e no shift happened .Therefore, in the case of the basic cosine function, f(x) = cos(x), the period is 2π
i.e
T = 2π
Therefore it will have frequency factor of b = 1
Option D: A sine function whose graph shows 2 cycles from -4pi radians to 0.
This statement is true .
2 cycles from -4pi radians to 0.
from (0,-2π) and from (-2π,-4π)
and Time period = 2π
Therefore it will have frequency factor of b = 1
Option E: A sine function whose graph shows 2 cycles from 0 to 2pi radians.
This statement is False .
2 cycles from 0 to 2pi
i.e (0,π) and (π,2π)
period = π
But sine period is = 2π
therefore it is false
Option F: A cosine function whose graph shows 4 cycles from 0 to 4pi radians.
This statement is False .
4 cycles from 0 to 4pi radians.
from (0,π ) , (π , 2π ),(2π , 3π )and (3π , 4π )
period = π
But cos period is = 2π
therefore it is false
Option G: A cosine function whose graph shows 1 cycle from 3pi radians to 5pi radians.
1 cycle from 3pi radians to 5pi radians.
period = 5π - 3π =2π
and cos period is also = 2π
therefore, this statement is true .
Hence, the frequency factor of b = 1 is of Option "A","C","D" and "G"
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