Question

Is anyone available to help with this?Find all solutions to the equation.7 sin^2x - 14 sin x + 2 = -5

Answer 1

We have the following trigonometric equation:

`7sin^2x-14sinx+2=-5`

1. Now, we can rewrite the equation as follows:

`\begin{gathered} 7sin^2x-14sinx=-5-2 \\ \\ 7sin^2x-14sinx=-7 \\ \\ \end{gathered}`

2. We have a common factor of 7 and sin(x). Then we have:

`\begin{gathered} 7(sin^2x-2sinx)=-7 \\ \\ 7sinx(sinx-2)=-7 \end{gathered}`

3. Now, we have:

`\begin{gathered} (7sinx)/(7)(sinx-2)=(-7)/(7) \\ \\ sinx(sinx-2)=-1 \\ \\ \end{gathered}`

4. Now, if we have:

`\begin{gathered} sinx=1 \\ \\ sinx-2=-1\Rightarrow sinx=-1+2=1 \\ \\ sinx=1 \end{gathered}`

5. Then, the solutions for this equation will be - applying the inverse function of the sine function to both sides of the equation:

`\begin{gathered} \sin^(-1)(sinx)=\sin^(-1)(1) \\ \\ x=(\pi)/(2)+2\pi n \end{gathered}`

Therefore, in summary, the values that are solutions for this equation are:

`x=(\pi)/(2)+2\pi n`

Where n is the any integer value.