Example 1
Which of the following graphs corresponds to the graph of $y=\sin(x+\dfrac{\pi}{4}) \cos(x+\dfrac{\pi}{4})$?
Solution
- One way to find the corresponding graph is to express the given function $y=\sin(x+\dfrac{\pi}{4}) \cos(x+\dfrac{\pi}{4})$ using a single function. Note that
$\sin (2x) = 2 \sin x \cos x$
Divide all terms of the above identity to obtain
$\sin x \cos x = \dfrac{1}{2} \sin(2x)$ - We apply the identity $\sin x \cos x = \dfrac{1}{2} \sin(2x)$ to rewrite the given function as follows
$y=\sin(x+\dfrac{\pi}{4}) \cos(x+\dfrac{\pi}{4})=\dfrac{1}{2} \sin(2(x+\dfrac{\pi}{4}))=\dfrac{1}{2} \sin(2x+\dfrac{\pi}{2})$
- We now evaluate $y$ at $x = 0$: $y(0) = \dfrac{1}{2} \sin(2(0)+\dfrac{\pi}{2})= \dfrac{1}{2} sin(\dfrac{\pi}{2}) = \dfrac{1}{2} \cdot 1 = \dfrac{1}{2}$
- Of all the 5 graphs only A and D are such that $y(0)=\dfrac{1}{2}$
- The period of $y = \dfrac{1}{2} \sin(2x+\dfrac{\pi}{2})$ is equal to $\dfrac{2\pi}{2}=\pi$. Of the graphs A and D, A has a period equal to $\pi$.
Answer A
