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How do you graph \[y=\sin \left( x+\dfrac{\pi }{2} \right)\]?

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Last updated date: 12th Jul 2024
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Answer
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Hint: In this problem, we have to graph the given trigonometric equation. We can use a trigonometric identity and we can simplify the given trigonometric expression to get its simplest form. Then we will get an equation for which we can plot the graph by assuming some values.

Complete step by step solution:
We know that the given trigonometric equation is,
 \[y=\sin \left( x+\dfrac{\pi }{2} \right)\]
We know that the trigonometric identity can be used in this problem is,
\[\sin \left( a+b \right)=\sin a\cos b+\cos a\sin b\]
We can now compare the above identity and the given trigonometric equation, we get
a = x, b = \[\dfrac{\pi }{2}\]
we can substitute these values in the trigonometric identity, we get
\[\Rightarrow y=\sin x\cos \left( \dfrac{\pi }{2} \right)+\cos x\sin \left( \dfrac{\pi }{2} \right)\]
Now we can substitute the trigonometric degree values in the above step for,
\[\begin{align}
  & \cos \dfrac{\pi }{2}=0 \\
 & \sin \dfrac{\pi }{2}=1 \\
\end{align}\]
we can substitute the above degree values, we get
\[\begin{align}
  & \Rightarrow y=\sin x\times 0+\cos x\times 1 \\
 & \Rightarrow y=\cos x \\
\end{align}\]
We can now find the x and y-intercept to plot the points
We know that at y-intercept the value of x is zero.
We can substitute x = 0, we get
\[\Rightarrow y=\cos \left( 0 \right)\]
We know that cos 0 = 1.
Therefore, the y-intercept is \[\left( 0,1 \right)\].
 We know that at x-intercept the value of y is zero.
We can substitute y = 0, we get
\[\Rightarrow 0=\cos x\]
We know that when x = \[\pm \dfrac{\pi }{2}\], then the value of y becomes 0.
Therefore, the x-intercepts are \[\left( \dfrac{\pi }{2},0 \right)\left( -\dfrac{\pi }{2},0 \right)\].
Now we can plot the graph
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Note: Students make mistakes while finding the correct degree values, which should be concentrated. We should know some trigonometric identities, formula, properties and degree values to solve these types of problems.