To find all angles [tex]\( 0^{\circ} \leq \theta < 360^{\circ} \)[/tex] that satisfy the equation [tex]\( \tan^2 \theta - 5 \tan \theta = 0 \)[/tex], we can follow these steps:
1. Rewriting the equation as a quadratic equation:
[tex]\[
\tan^2 \theta - 5 \tan \theta = 0
\][/tex]
This equation can be factored to:
[tex]\[
\tan \theta (\tan \theta - 5) = 0
\][/tex]
2. Solving for [tex]\( \tan \theta \)[/tex]:
[tex]\[
\tan \theta = 0 \quad \text{or} \quad \tan \theta - 5 = 0
\][/tex]
This results in:
[tex]\[
\tan \theta = 0 \quad \text{or} \quad \tan \theta = 5
\][/tex]
3. Finding angles for [tex]\(\tan \theta = 0\)[/tex]:
- The tangent function is zero at angles [tex]\( \theta = 0^\circ \)[/tex] and [tex]\( \theta = 180^\circ \)[/tex] within the given range [tex]\( 0^\circ \leq \theta < 360^\circ \)[/tex].
4. Finding angles for [tex]\(\tan \theta = 5\)[/tex]:
- We need to find [tex]\( \theta \)[/tex] such that the tangent of [tex]\( \theta \)[/tex] is 5. Such angles occur at:
[tex]\[
\theta = \arctan(5)
\][/tex]
- Calculating the principal value, we have:
[tex]\[
\theta \approx 78.7^\circ
\][/tex]
- The tangent function repeats every [tex]\( 180^\circ \)[/tex] because it is periodic with a period of [tex]\( 180^\circ \)[/tex]. Therefore, the other angle within the range is:
[tex]\[
\theta = 78.7^\circ + 180^\circ = 258.7^\circ
\][/tex]
5. Combining all valid angles:
- For [tex]\( \tan \theta = 0 \)[/tex], the angles are [tex]\( 0^\circ \)[/tex] and [tex]\( 180^\circ \)[/tex].
- For [tex]\( \tan \theta = 5 \)[/tex], the angles are [tex]\( 78.7^\circ \)[/tex] and [tex]\( 258.7^\circ \)[/tex].
Therefore, the angles [tex]\( \theta \)[/tex] that satisfy the equation [tex]\( \tan^2 \theta - 5 \tan \theta = 0 \)[/tex] within the range [tex]\( 0^\circ \leq \theta < 360^\circ \)[/tex] are:
[tex]\[ \boxed{0.0^\circ, 180.0^\circ, 78.7^\circ, 258.7^\circ} \][/tex]
