Question

If $\cos x=k$ has exactly one solution in $\left[ 0,2\pi \right]$ , then write the value(s) of k.

Answer 1

Hint: The equations that involve the trigonometric functions of a variable are called trigonometric equations. We will try to find the solutions of such equations. These equations have one or more trigonometric ratios of unknown angles.

Complete step-by-step answer:

Let us consider the value of k is 0, 1, and -1.

The given trigonometric equation is $\cos x=k$

Case I: If k = 0 , then

$\cos x=0$

The general solution of the trigonometric equation \[\cos \theta =0\] is $\theta =\left( 2n+1 \right)\dfrac{\pi }{2},n\in Z$

$x=\left( 2n+1 \right)\dfrac{\pi }{2},n\in Z$

Now, for the values of n = 1, 2, 3,…………….

$x=\dfrac{3\pi }{2},\dfrac{5\pi }{2},\dfrac{7\pi }{2}...............$

Case II If k = 1, then

$\cos x=1$

We know that $\cos 0=1$

$\cos x=\cos 0$

The general solution of the trigonometric equation \[\cos \theta =\cos 0\]is $\theta =2m\pi ,m\in Z$

$x=2m\pi ,m\in Z$

Now, for the values of m = 1, 2, 3,…………….

$x=2\pi ,4\pi ,6\pi ...................$

Case III: If k = -1 then

$\cos x=-1$

We know that $\cos x=\cos \pi $

$\cos x=\cos \pi $

The general solution of the trigonometric equation \[\cos \theta =\cos \pi \]is $\theta =2p\pi \pm \alpha ,p\in Z$

$\theta =2p\pi \pm \pi ,p\in Z$

Now, for the values of p = 1, 2, 3,…………….

$x=2p\pi +\pi =3\pi ,5\pi ,7\pi ...................$

And

$x=2p\pi -\pi =\pi ,3\pi ,5\pi ...................$

Clearly, we can see that for $x=\pi $ lies between 0 and $2\pi $ .

Hence $\cos x=k$ has exactly one solution.

Therefore k =-1

Note: The solution of a trigonometric equation of an unknown angle $\theta $ , where $0\le \theta \le 2\pi $ are called as its principal solutions. For example- If $\sin \theta =\dfrac{1}{2}$, then $\theta =\dfrac{\pi }{6},\dfrac{5\pi }{6}$ are its principal solutions.