Sketch the graphs of each of the following for $x$ in [0,2$\pi$]. list the $x$-axis intercepts of each graph for this interval.
$$y=\sqrt{2} \cos \left(x-\frac{\pi}{4}\right)+1$$
I tried to solve the equation $y = 0$ by performing the following steps:
$$\begin{align*}
-1&=\sqrt{2} \cos\left(x-\frac{\pi}{4}\right) \\\\
-\frac{1}{\sqrt{2}}&=\cos\left(x-\frac{\pi}{4}\right) \\\\
-\frac{1}{\sqrt{2}}+\frac{\pi}{4}&=\cos(x) \end{align*}$$
I'm unable to proceed further. Can you please explain this to me in a step-by-step fashion? And I think that after you've got the answer you have to look the unit circle to get the exact answer. Please also show me how to look for the answer from unit cirle.
Thanks so much!
Best Answer
As mentioned in the comments, you can't add $\frac{\pi}{4}$ to both sides, you can think of it as being "trapped" inside the cosine. Your work is correct up to your second step: \begin{align*} -1&=\sqrt{2} \cos\left(x-\frac{\pi}{4}\right) \\\\ -\frac{1}{\sqrt{2}}&=\cos\left(x-\frac{\pi}{4}\right) \end{align*} Now, this is when you look at the unit circle. You will notice that $\cos(x) = -\frac{1}{\sqrt{2}}$ when $x = \frac{3\pi}{4}$ and $x = \frac{5\pi}{4}$. This means that we have to solve the following equations: $$ x-\frac{\pi}{4} = \frac{3\pi}{4} \quad \quad \mbox{and} \quad \quad x-\frac{\pi}{4} = \frac{5\pi}{4}. $$ Doing this gives you $x = \pi$ and $x = \frac{3\pi}{2}$.