Chapter 5, Problem 42RE

Solution

To calculate: The solutions of the equation $\cos 2t=\cos t$ in the interval $\left[0,2\pi \right)$ .

The solutions of the equation $\cos 2t=\cos t$ interval $\left[0,2\pi \right)$ are $x=\left\{0,\frac{2\pi }{3},\frac{4\pi }{3}\right\}$ .

Given information: The given equation is $\cos 2t=\cos t$ .

Calculation:

Apply double-angle identity and solve the given equation to find the value of $x$ .

  $\begin{array}{c}\cos 2t=\cos t\\ 2{\cos }^{2}t-1=\cos t\\ 2{\cos }^{2}t-\cos t-1=0\\ 2{\cos }^{2}t-2\cos t+\cos t-1=0\end{array}$

Further simplify the equation.

  $\left(2\cos t+1\right)\left(\cos t-1\right)=0$

Equate the first factor to zero.

  $\begin{array}{c}2\cos t+1=0\\ 2\cos t=-1\\ \cos t=-\frac{1}{2}\\ t=\frac{2\pi }{3},\frac{4\pi }{3}\end{array}$

Equate the second factor to zero.

  $\begin{array}{c}\cos t-1=0\\ \cos t=1\\ t={\cos }^{-1}\left(1\right)\\ t=0\end{array}$

Thus, the solutions of the equation $\cos 2t=\cos t$ interval $\left[0,2\pi \right)$ are $x=\left\{0,\frac{2\pi }{3},\frac{4\pi }{3}\right\}$ .