Question

solve each equation on the interval $0^{circ},360^{circ})$. solving trigonometric equations using substitutio...i
$\sin ( \frac { x } { 2 } ) = \sqrt { 2 } - \sin ( \frac { x } { 2 } )$
$u = \frac { x } { 2 } \
ightarrow x = 2 u$
$\sin ( u ) = \sqrt { 2 } - \sin ( u )$
$\frac { 2 \sin ( u ) } { 2 } = \frac { \sqrt { 2 } } { 2 }$
$\sin ( u ) = \frac { \sqrt { 2 } } { 2 }$
$u = 45 ^ { \circ } \
ightarrow x _ { 1 } = 2 ( 45 ^ { \circ } ) = 90 ^ { \circ }$
$x _ { 2 } = 2 ( 135 ^ { \circ } ) = 270 ^ { \circ }$
solve $3 \cos ( 4 x ) = 2$ for the smallest positive solution.
give your answer accurate to at least two decimal places.

Explanation:

Step1: Isolate the cosine function

Divide both sides of the equation \(3\cos(4x)=2\) by \(3\):
\(\cos(4x)=\frac{2}{3}\)

Step2: Solve for \(4x\)

Take the inverse cosine of both sides. Since we want the smallest positive solution, we use \(4x = \cos^{-1}(\frac{2}{3})\)
Using a calculator, \(\cos^{-1}(\frac{2}{3})\approx48.19^{\circ}\) (in degree mode)

Step3: Solve for \(x\)

Divide both sides of \(4x\approx48.19^{\circ}\) by \(4\)
\(x=\frac{48.19^{\circ}}{4}=12.05^{\circ}\)

Answer:

\(12.05\)