STEP - BY - STEP EXPLANATION
What to do?
Solve the given trigonometric equation.
Given:
[tex]12\sin ^2x-20\sin x=-8[/tex]To solve the given problem, we will follow the steps below:
Step 1
Let y=sinx
Step 2
Replace sinx by y.
[tex]12y^2-20y=-8[/tex]Step 2
Re-arrange the above into the form:
[tex]ax^2+bx+c=0_{}[/tex]That is;
[tex]12y^2-20y+8=0[/tex]a=12 b=-20 and c=8
Step 3
Use the quadratic formula below to solve.
[tex]y=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex][tex]y=\frac{20\pm\sqrt[]{(-20)^2-4(12)(8)}}{2(12)}[/tex][tex]=\frac{20\pm\sqrt[]{400-384}}{24}[/tex][tex]\begin{gathered} =\frac{20\pm\sqrt[]{16}}{24} \\ \\ =\frac{20\pm4}{24} \end{gathered}[/tex]Either
[tex]\begin{gathered} y=\frac{20+4}{24}=\frac{24}{24}=1 \\ \\ or \\ \\ y=\frac{20-4}{20}=\frac{16}{24}=\frac{2}{3} \end{gathered}[/tex]Step 4
Substitute the value of y in;
y=sinx
If y = 1
Then,
[tex]\sin x=1[/tex][tex]x=\sin ^{-1}(1)[/tex][tex]x=90\degree[/tex][tex]\begin{gathered} x=90\times\frac{\pi}{180} \\ \\ =\frac{\pi}{2}\text{rads} \end{gathered}[/tex]Step 5
Substitute y=2/3 in y=sinx
[tex]\sin x=\frac{2}{3}[/tex][tex]x=\sin ^{-1}(\frac{2}{3})[/tex][tex]x=41.8103148958\degree[/tex]Change to radians
[tex]x=41.8810348958\times\frac{\pi}{180}[/tex][tex]=0.2323\pi rads[/tex]Therefore,
[tex]x=\frac{1}{2}\pi rads,0.2323\pi rads[/tex]