Chapter 13.1, Problem 61E

To determine

To calculate: The value of x for the equation $\sin 2x+2\sin x=0$ .

Answer to Problem 61E

The value of x for the equation $\sin 2x+2\sin x=0$ are $0°,180°\text{ and }360°$ .

Explanation of Solution

Given information:

The equation $\sin 2x+2\sin x=0$ and range of xas $0°\le x\le 360°$ .

Formula used:

The double angle indent for sine trigonometric function is $\sin 2x=2\sin x\cos x$ .

Calculation:

Consider the expression equation $\sin 2x+2\sin x=0$ .

Recall double angle indent for sine trigonometric function is $\sin 2x=2\sin x\cos x$ .

Rewrite the equation as,

  $\begin{array}{c}\sin 2x+2\sin x=0\\ 2\sin x\cos x+2\sin x=0\\ 2\sin x\left(\cos x+1\right)=0\end{array}$

Apply the zero property. Either $2\sin x=0\text{ or }\cos x+1=0$ .

That is $\sin x=0$ or $\cos x=-1$ .

The range of xis $0°\le x\le 360°$ .

In the provided range is sine function is zero at $0°$ and at every multiple of $180°$ .

That is $\sin x=0$ at $x=0°,x=180°\text{ and }x=360°$ .

In the provided range is cosine function is $-1$ at $180°$ .

That is $\cos x=-1$ at $x=180°$ .

Therefore, the values are $x=0°,x=180°\text{ and }x=360°$ .

Thus, the value of x for the equation $\sin 2x+2\sin x=0$ are $0°,180°\text{ and }360°$ .