Answer
Verified
375.3k+ views
Hint: The given equation is a sine wave equation. The standard form of sine function is $ f\left( x \right) = a\sin \left( {b\left( {x - c} \right)} \right) $ . The amplitude of a sine wave is given by $ \left| a \right| $ , the period is given by $ \dfrac{{2\pi }}{{\left| b \right|}} $ and the phase shift is given by c . We are going to compare the given equation with the standard form and then obtain the values of a, b and c.
Complete step-by-step answer:
In this question, we are given a function of sine wave and we are supposed to find the amplitude, period and the phase shift of this function.
First of all, what is sine wave?
A sine wave is a geometric waveform that oscillates periodically and is defined by the function
$ y = \sin x $ .
The standard form of sine function is given by:
$ \Rightarrow f\left( x \right) = a\sin \left( {b\left( {x - c} \right)} \right) $ - - - - - - - (1)
Now, this sine function has some properties. Let us discuss these properties.
$ \to $ Amplitude: Amplitude is the vertical distance between the sinusoidal axis and maximum or minimum value of the function.
The amplitude of this function is given by $ \left| a \right| $ .
$ \to $ Period: Period is the length of one cycle of the curve.
The period of this function is given by $ \dfrac{{2\pi }}{{\left| b \right|}} $ .
$ \to $ Phase shift: Phase shift is the angle in degrees or radian that the waveform has shifted from a certain point along the horizontal zero axis.
The phase shift of this function is given by $ c $ units to the right.
So, now know everything that we need to find.
Therefore, $ f\left( x \right) = 3\sin \left( {2x + \pi } \right) $
To get the value of b, take 2 common out, we get
$ f\left( x \right) = 3\sin \left( {2\left( {x + \dfrac{\pi }{2}} \right)} \right) $
Now, compare $ f\left( x \right) = 3\sin \left( {2\left( {x + \dfrac{\pi }{2}} \right)} \right) $ with equation $ f\left( x \right) = a\sin \left( {b\left( {x - c} \right)} \right) $ , we will get the values of a, b and c.
Therefore, $ a = 3,b = 2,c = - \dfrac{\pi }{2} $ .
Now, we can find the amplitude, period, phase shift easily.
$ \to $ Amplitude $ = \left| a \right| = \left| 3 \right| = 3 $
$ \to $ Period $ = \dfrac{{2\pi }}{{\left| b \right|}} = \dfrac{{2\pi }}{{\left| 2 \right|}} = \dfrac{{2\pi }}{2} = \pi $
$ \to $ Phase shift $ = c = - \dfrac{\pi }{2} $
Note: If the value of phase shift is negative, the sine wave will shift in the right direction and if the value of the phase shift is positive, the sine wave will be shifted in the left direction. In our question, we got phase shift as $ - \dfrac{\pi }{2} $ so, the sine wave will be shifted in the right direction.
Complete step-by-step answer:
In this question, we are given a function of sine wave and we are supposed to find the amplitude, period and the phase shift of this function.
First of all, what is sine wave?
A sine wave is a geometric waveform that oscillates periodically and is defined by the function
$ y = \sin x $ .
The standard form of sine function is given by:
$ \Rightarrow f\left( x \right) = a\sin \left( {b\left( {x - c} \right)} \right) $ - - - - - - - (1)
Now, this sine function has some properties. Let us discuss these properties.
$ \to $ Amplitude: Amplitude is the vertical distance between the sinusoidal axis and maximum or minimum value of the function.
The amplitude of this function is given by $ \left| a \right| $ .
$ \to $ Period: Period is the length of one cycle of the curve.
The period of this function is given by $ \dfrac{{2\pi }}{{\left| b \right|}} $ .
$ \to $ Phase shift: Phase shift is the angle in degrees or radian that the waveform has shifted from a certain point along the horizontal zero axis.
The phase shift of this function is given by $ c $ units to the right.
So, now know everything that we need to find.
Therefore, $ f\left( x \right) = 3\sin \left( {2x + \pi } \right) $
To get the value of b, take 2 common out, we get
$ f\left( x \right) = 3\sin \left( {2\left( {x + \dfrac{\pi }{2}} \right)} \right) $
Now, compare $ f\left( x \right) = 3\sin \left( {2\left( {x + \dfrac{\pi }{2}} \right)} \right) $ with equation $ f\left( x \right) = a\sin \left( {b\left( {x - c} \right)} \right) $ , we will get the values of a, b and c.
Therefore, $ a = 3,b = 2,c = - \dfrac{\pi }{2} $ .
Now, we can find the amplitude, period, phase shift easily.
$ \to $ Amplitude $ = \left| a \right| = \left| 3 \right| = 3 $
$ \to $ Period $ = \dfrac{{2\pi }}{{\left| b \right|}} = \dfrac{{2\pi }}{{\left| 2 \right|}} = \dfrac{{2\pi }}{2} = \pi $
$ \to $ Phase shift $ = c = - \dfrac{\pi }{2} $
Note: If the value of phase shift is negative, the sine wave will shift in the right direction and if the value of the phase shift is positive, the sine wave will be shifted in the left direction. In our question, we got phase shift as $ - \dfrac{\pi }{2} $ so, the sine wave will be shifted in the right direction.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
10 examples of evaporation in daily life with explanations
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Why is there a time difference of about 5 hours between class 10 social science CBSE