Chapter 5 - Section 5.5 - Trigonometric Equations - Exercise Set - Page 703: 28

The solutions in the interval $[0,2\pi )$ are $\frac{5\pi }{16}$, $\frac{7\pi }{16}$, $\frac{13\pi }{16}$, $\frac{15\pi }{16}$, $\frac{21\pi }{16}$, $\frac{23\pi }{16}$, $\frac{29\pi }{16}$ and $\frac{31\pi }{16}$.

We know that the period of the sine function is $2\pi $. In the interval there are two values at which the sine function is $-\frac{\sqrt{2}}{2}$. One is $\pi +\frac{\pi }{4}=\frac{5\pi }{4}$ and since sine is negative in quadrant IV, the other value is: $\begin{align} & 2\pi -\frac{\pi }{4}=\frac{8\pi -\pi }{4} \\ & =\frac{7\pi }{4} \end{align}$ Therefore, all the solutions to $\sin 4x=-\frac{\sqrt{2}}{2}$ are given by: $\begin{align} & 4x=\frac{5\pi }{4}+2n\pi \\ & x=\frac{5\pi }{16}+\frac{n\pi }{2} \end{align}$ Or, $\begin{align} & 4x=\frac{7\pi }{4}+2n\pi \\ & x=\frac{7\pi }{16}+\frac{n\pi }{2} \end{align}$ Where n is any integer. The solutions in the interval $[0,2\pi )$ are obtained by letting $n=0$, $n=1$, $n=2$, and $n=3$. The equation is calculated by taking first $n$ as 0 and then as 1, 2, and 3. It can be further simplified as follows. $\begin{align} & x=\frac{5\pi }{16}+\frac{n\pi }{2} \\ & =\frac{5\pi }{16}+\frac{0\times \pi }{2} \\ & =\frac{5\pi }{16}+0 \\ & =\frac{5\pi }{16} \end{align}$ $\begin{align} & x=\frac{5\pi }{16}+\frac{n\pi }{2} \\ & =\frac{5\pi }{16}+\frac{1\times \pi }{2} \\ & =\frac{5\pi }{16}+\frac{1\pi }{2} \\ & =\frac{5\pi +8\pi }{16} \end{align}$ $=\frac{13\pi }{16}$ $\begin{align} & x=\frac{5\pi }{16}+\frac{n\pi }{2} \\ & =\frac{5\pi }{16}+\frac{2\times \pi }{2} \\ & =\frac{5\pi }{16}+\frac{2\pi }{2} \\ & =\frac{5\pi +16\pi }{16} \end{align}$ $=\frac{21\pi }{16}$ $\begin{align} & x=\frac{5\pi }{16}+\frac{n\pi }{2} \\ & =\frac{5\pi }{16}+\frac{3\times \pi }{2} \\ & =\frac{5\pi }{16}+\frac{3\pi }{2} \\ & =\frac{5\pi +24\pi }{16} \end{align}$ $=\frac{29\pi }{16}$ The second equation is also computed by taking first $n$ as 0 and then as 1, 2, and 3. It can be further simplified as follows. $\begin{align} & x=\frac{7\pi }{16}+\frac{n\pi }{2} \\ & =\frac{7\pi }{16}+\frac{0\times \pi }{2} \\ & =\frac{7\pi }{16}+0 \\ & =\frac{7\pi }{16} \end{align}$ $\begin{align} & x=\frac{7\pi }{16}+\frac{n\pi }{2} \\ & =\frac{7\pi }{16}+\frac{1\times \pi }{2} \\ & =\frac{7\pi }{16}+\frac{\pi }{2} \\ & =\frac{7\pi +8}{16} \end{align}$ $=\frac{15\pi }{16}$ $\begin{align} & x=\frac{7\pi }{16}+\frac{n\pi }{2} \\ & =\frac{7\pi }{16}+\frac{2\times \pi }{2} \\ & =\frac{7\pi }{16}+\frac{2\pi }{2} \\ & =\frac{7\pi +16}{16} \end{align}$ $=\frac{23\pi }{16}$ $\begin{align} & x=\frac{7\pi }{16}+\frac{n\pi }{2} \\ & =\frac{7\pi }{16}+\frac{3\times \pi }{2} \\ & =\frac{7\pi }{16}+\frac{3\pi }{2} \\ & =\frac{7\pi +24\pi }{16} \end{align}$ $=\frac{31\pi }{16}$