Chapter 7.5, Problem 23E

(a)

To determine

To find: The solution of the equation $\cos \frac{\theta }{2}-1=0$ .

Answer to Problem 23E

The solutions of the equation $\cos \frac{\theta }{2}-1=0$ are $\theta =2k\pi$ .

Explanation of Solution

Given information:

The given equation is $\cos \frac{\theta }{2}-1=0$ .

Calculation:

Simplify the equation.

  $\begin{array}{c}\cos \frac{\theta }{2}-1=0\\ \cos \frac{\theta }{2}=1\end{array}$

As cosine has period $2\pi$ . So first find the solutions in $\left[0,2\pi \right)$ .

Therefore, the solution to the equation in $\left[0,2\pi \right)$ are

  $\begin{array}{c}\frac{\theta }{2}={\cos }^{-1}\left(1\right)\\ =0,2\pi \end{array}$

As cosine repeats its values after every $2\pi$ units.

So, the general solution of the equation is given by

  $\begin{array}{c}\frac{\theta }{2}=0+2k\pi ,2\pi +2k\pi \\ \frac{\theta }{2}=0+2k\pi \\ \theta =4k\pi \end{array}$

Therefore, the solutions of the equation $\cos \frac{\theta }{2}-1=0$ are $\theta =4k\pi$ .

(b)

To determine

To find: The solution of the equation $\cos \frac{\theta }{2}-1=0$ in $\left[0,2\pi \right)$ .

Answer to Problem 23E

The solution of the equation $\cos \frac{\theta }{2}-1=0$ in $\left[0,2\pi \right)$ is $\theta =0$ .

Explanation of Solution

Given information:

The given equation is $\cos \frac{\theta }{2}-1=0$ and the interval is $\left[0,2\pi \right)$ .

Calculation:

Simplify the equation.

  $\begin{array}{c}\cos \frac{\theta }{2}-1=0\\ \cos \frac{\theta }{2}=1\end{array}$

Therefore, the solution to the equation in $\left[0,2\pi \right)$ are

  $\begin{array}{c}\frac{\theta }{2}={\cos }^{-1}\left(1\right)\\ \frac{\theta }{2}=0,\pi \\ \theta =0\end{array}$

Therefore, the solution of the equation $\cos \frac{\theta }{2}-1=0$ in $\left[0,2\pi \right)$ is $\theta =0$ .