Given the expression:
[tex]12sin^2(x)-3=0[/tex]
To solve the expression, follow the steps below:
Step 01: Add 3 to both sides.
[tex]\begin{gathered} 12sin^2(x)-3+3=0+3 \\ 12sin^2(x)=3 \end{gathered}[/tex]
Step 02: Divide both sides by 12.
[tex]\begin{gathered} \frac{12sin^2(x)}{12}=\frac{3}{12} \\ sin^2(x)=\frac{1}{4} \end{gathered}[/tex]
Step 03: Take the square root of both sides.
[tex]\begin{gathered} \sqrt{sin^2(x)}=\pm\sqrt{\frac{1}{4}} \\ sin(x)=\pm\frac{1}{2} \end{gathered}[/tex]
Step 04: Evaluate the results.
First, let's evaluate sin(x) = 1/2.
Sin(x) is positive in the first and in the second quadrant. Then,
[tex]\begin{gathered} sin^{-1}(\frac{1}{2})=x \\ x=\frac{\pi}{6},x=\frac{5}{6}\pi \end{gathered}[/tex]
Second, let's evaluate sin(x) = -1/2.
Sin(x) is negative in the third and in fourth quadrant. Then,
[tex]\begin{gathered} sin^{-1}(-\frac{1}{2})=x \\ x=\frac{7}{6}\pi,x=\frac{11}{6}\pi \end{gathered}[/tex]
Answer:
[tex]x=\frac{\pi}{6},x=\frac{5}{6}\pi,x=\frac{7}{6}\pi,x=\frac{11}{6}\pi[/tex]
