4 months ago
Q:

Find the exact solution [0,2pi) of 8sin^2(x)+6 sin (x)+1=0

Accepted Solution

A:

Solution variant #2.

To get an equation that is easier to solve, substitute $t$ for $\sin\left({x}\right)$
$8{t}^{2}+6t+1=0$
Solve the equation for $t$
$\begin{array} { l }t=-\frac{ 1 }{ 2 },\\t=-\frac{ 1 }{ 4 }\end{array}$
Substitute back $t=\sin\left({x}\right)$
$\begin{array} { l }\sin\left({x}\right)=-\frac{ 1 }{ 2 },\\\sin\left({x}\right)=-\frac{ 1 }{ 4 }\end{array}$
Solve the equation for $x$
$\begin{array}{l}\begin{array}{l}x={210}\degree \\ \end{array}, \\ \begin{array}{l}x={330}\degree \\ \end{array}, \\ \sin\left({x}\right)=-\frac{ 1 }{ 4 }\end{array}$
Solve the equation for $x$
$\begin{array}{l}\begin{array}{l}x={210}\degree \\ \end{array}, \\ \begin{array}{l}x={330}^o \\ \end{array}, \\ \\ \begin{array}{l}x={180}\degree+\arcsin\left({\frac{ 1 }{ 4 }}\right) \\ \end{array}\end{array}$
Find the approximate value
$\begin{array}{l}\begin{array}{l}x={210}\degree \\ \end{array}, \\ \begin{array}{l}x={330}\degree \\ \end{array} \\ \begin{array}{l}x\approx-{345.5}\degree \\ \end{array}, \\ \begin{array}{l}x\approx{194.5}^o \\ \end{array}\end{array}$