Chapter C, Problem 29E

To determine

To calculate:

The value of $x$ to satisfy the following equation:

  $2\cos x-1=0$

Answer to Problem 29E

The values of $x$ for the interval $\left[0,2\pi \right]$ to satisfy eq. $2\cos x-1=0$ are $\frac{\pi }{3},\frac{5\pi }{3}$ .

Explanation of Solution

Given information:

  $\begin{array}{l}2\cos x-1=0\\ \left[0,2\pi \right]\end{array}$

Calculation:

Know that:

  $\begin{array}{l}2\cos x-1=0\\ 2\cos x=1\\ \cos x=\frac{1}{2}\end{array}$

Now, consider the values of $x$ in the given interval $\left[0,2\pi \right]$ that cause cosine value is equal to $\frac{1}{2}$ .

Now, $\cos x=\frac{1}{2}$ is positive that means it should be in the first and fourth quadrants.

Know that:

The circle has co-ordinates $\left(x,y\right)=\left(\cos \theta ,\sin \theta \right)$ with the radius $1$ .

Now, the angle corresponding to the $x-$ co-ordinates means $\cos x=\frac{1}{2}$ , a unit circle is drawn as:

  

Fig. Unit circle of $\cos x=\frac{1}{2}$

From the diagram:

Cosine is equal to $\frac{1}{2}$ when the angle is $\frac{\pi }{3},\frac{5\pi }{3}$ .

Therefore, the values of $x$ for the interval $\left[0,2\pi \right]$ to satisfy eq. $2\cos x-1=0$ are $\frac{\pi }{3},\frac{5\pi }{3}$ .