We have the following trigonometric equation:
[tex]7sin^2x-14sinx+2=-5[/tex]
1. Now, we can rewrite the equation as follows:
[tex]\begin{gathered} 7sin^2x-14sinx=-5-2 \\ \\ 7sin^2x-14sinx=-7 \\ \\ \end{gathered}[/tex]
2. We have a common factor of 7 and sin(x). Then we have:
[tex]\begin{gathered} 7(sin^2x-2sinx)=-7 \\ \\ 7sinx(sinx-2)=-7 \end{gathered}[/tex]
3. Now, we have:
[tex]\begin{gathered} \frac{7sinx}{7}(sinx-2)=\frac{-7}{7} \\ \\ sinx(sinx-2)=-1 \\ \\ \end{gathered}[/tex]
4. Now, if we have:
[tex]\begin{gathered} sinx=1 \\ \\ sinx-2=-1\Rightarrow sinx=-1+2=1 \\ \\ sinx=1 \end{gathered}[/tex]
5. Then, the solutions for this equation will be - applying the inverse function of the sine function to both sides of the equation:
[tex]\begin{gathered} \sin^{-1}(sinx)=\sin^{-1}(1) \\ \\ x=\frac{\pi}{2}+2\pi n \end{gathered}[/tex]
Therefore, in summary, the values that are solutions for this equation are:
[tex]x=\frac{\pi}{2}+2\pi n[/tex]
Where n is the any integer value.
