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Solve the equation for exact solutions in the interval [tex]\left[0^{\circ}, 360^{\circ}\right)[/tex]. Use an algebraic method.

[tex]
9 \sec^2 \theta \tan \theta = 12 \tan \theta
[/tex]


Sagot :

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]
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