To solve the given equation [tex]\( 9 \sec^2 \theta \tan \theta = 12 \tan \theta \)[/tex] within the interval [tex]\([0^\circ, 360^\circ)\)[/tex], follow these steps:
1. Simplify the Equation:
Begin by moving all terms to one side of the equation:
[tex]\[
9 \sec^2 \theta \tan \theta - 12 \tan \theta = 0
\][/tex]
2. Factor Out Common Terms:
Factor out [tex]\(\tan \theta\)[/tex] from the equation:
[tex]\[
\tan \theta (9 \sec^2 \theta - 12) = 0
\][/tex]
3. Solve Each Factor:
This produces two equations to solve:
[tex]\[
\tan \theta = 0
\][/tex]
[tex]\[
9 \sec^2 \theta - 12 = 0
\][/tex]
4. Solve for [tex]\(\tan \theta = 0\)[/tex]:
The tangent function is zero at angles where [tex]\(\theta = k \cdot 180^\circ\)[/tex] for any integer [tex]\(k\)[/tex]. In the interval [tex]\([0^\circ, 360^\circ)\)[/tex], the solutions are:
[tex]\[
\theta = 0^\circ, \; 180^\circ
\][/tex]
5. Solve for [tex]\(9 \sec^2 \theta - 12 = 0\)[/tex]:
Convert [tex]\(\sec^2 \theta\)[/tex] to [tex]\(1 + \tan^2 \theta\)[/tex]:
[tex]\[
9 ( 1 + \tan^2 \theta ) - 12 = 0
\][/tex]
Simplify to find [tex]\(\tan^2 \theta\)[/tex]:
[tex]\[
9 + 9 \tan^2 \theta - 12 = 0
\][/tex]
[tex]\[
9 \tan^2 \theta = 3
\][/tex]
[tex]\[
\tan^2 \theta = \frac{1}{3}
\][/tex]
Take the square root of both sides:
[tex]\[
\tan \theta = \pm \frac{1}{\sqrt{3}} = \pm \frac{\sqrt{3}}{3}
\][/tex]
6. Find Angles Corresponding to [tex]\(\tan \theta = \frac{\sqrt{3}}{3}\)[/tex]:
Recognizing the values of tangent, we have [tex]\(\theta\)[/tex] at:
[tex]\( \theta = 30^\circ\)[/tex] and [tex]\( \theta = 210^\circ \)[/tex] where [tex]\(\tan \theta = \frac{\sqrt{3}}{3}\)[/tex],
[tex]\( \theta = 150^\circ \)[/tex] and [tex]\( \theta = 330^\circ \)[/tex] where [tex]\(\tan \theta = -\frac{\sqrt{3}}{3}\)[/tex].
7. Combine All Solutions:
Collect all unique solutions in the interval [tex]\([0^\circ, 360^\circ)\)[/tex]:
[tex]\[
\theta = 0^\circ, 30^\circ, 150^\circ, 180^\circ, 210^\circ, 330^\circ
\][/tex]
Hence, the exact solutions to the equation [tex]\( 9 \sec^2 \theta \tan \theta = 12 \tan \theta \)[/tex] within the interval [tex]\([0^\circ, 360^\circ)\)[/tex] are:
[tex]\[
0^\circ, 30^\circ, 150^\circ, 180^\circ, 210^\circ, 330^\circ
\][/tex]
