Question

How do you find phase shift in a trigonometric function $y=\csc \left( 2\theta +\pi \right)-3$?

Answer 1

From the given question, we have been asked to find phase shift in a trigonometric function $y=\csc \left( 2\theta +\pi \right)-3$.Phase shift is the horizontal shift left or right for periodic functions. The general form is $y=A\csc \left( B\theta +C \right)+D$ where the time period is $T=\dfrac{2\pi }{B}$ and phase shift $\phi =\dfrac{-C}{B}$ .

Complete step-by-step solution:
First of all, we have to write the general form of a cosine function to find the phase shift,
The general form: $y=A\csc \left( B\theta +C \right)+D$ where $A$ in the general form represents amplitude.
And also, the period is $T=\dfrac{2\pi }{B}$
The phase shift, $\phi =-\dfrac{C}{B}$
The vertical shift is $D$.
Now, from the given question we have been asked to find the phase shift in a trigonometric function $y=\csc \left( 2\theta +\pi \right)-3$
The given question is in the general form of a cosine function.
Now, what we have to do is compare the coefficients in the general form which we have discussed above and the given question.
By comparing the coefficients in general form and the given question, we get the below values,
$B=2$
$c=\pi$
Now, in the question we have been asked to find the phase shift.
By the above written general form, we got the formula for phase shift. As we already got the values, substitute those values in the formula to get the phase shift.
Phase shift $\phi =-\dfrac{C}{B}$
Therefore $\phi =-\dfrac{\pi }{2}$
Hence, we got the phase shift for the given function $y=\csc \left( 2\theta +\pi \right)-3$ as $\phi =-\dfrac{\pi }{2}$.

Note: We should be well aware of the concept of phase shift in a trigonometric function. We should be well aware of the general form to know the formula for the phase shift of a cosine function. We should be very careful while comparing the coefficients of the general form and the given question. Values must be substituted exactly and correctly to get the correct answer for the given function. Similarly the general equation for sine trigonometric function is $y=A\sin \left( \theta B+C \right)+D$ where the time period is $T=\dfrac{2\pi }{B}$ and phase shift is $\phi =\dfrac{-C}{B}$ and the vertical shift is $\D$.