Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree. $\newline$ $6 \tan ^{2} \theta=11 \tan \theta+7$ $\newline$ Answer: $\theta=$

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Algebra 2

Csc, sec, and cot of special angles

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Q. Find all angles,

0^{\circ} \leq \theta<360^{\circ}

, that satisfy the equation below, to the nearest tenth of a degree.

\newline

6 \tan ^{2} \theta=11 \tan \theta+7

\newline

Answer:

\theta=

Move to Quadratic Form: Write the given equation in a quadratic form by moving all terms to one side. $\newline$ $6\tan^2(\theta) - 11\tan(\theta) - 7 = 0$
Factor and Solve: Factor the quadratic equation to find the values of $\tan(\theta)$ . $(3\tan(\theta) + 2)(2\tan(\theta) - 7) = 0$
Solve for $\tan(\theta)$ : Set each factor equal to zero and solve for $\tan(\theta)$ . $3\tan(\theta) + 2 = 0$ or $2\tan(\theta) - 7 = 0$
Calculate Angles for $-\frac{2}{3}$ : Solve the first equation for $\tan(\theta)$ . $\newline$ $\tan(\theta) = -\frac{2}{3}$
Calculate Angles for $\frac{7}{2}$ : Solve the second equation for $\tan(\theta)$ . $\newline$ $\tan(\theta) = \frac{7}{2}$
Calculate Angles for $7/2$ : Solve the second equation for $\tan(\theta)$ . $\newline$ $\tan(\theta) = \frac{7}{2}$ Find the angles that correspond to $\tan(\theta) = -\frac{2}{3}$ in the range 0^\circ \leq \theta < 360^\circ. $\newline$ Using a calculator, we find that $\theta \approx 180^\circ - \arctan\left(\frac{2}{3}\right)$ and $\theta \approx 360^\circ - \arctan\left(\frac{2}{3}\right)$ .
Calculate Angles for $\frac{7}{2}$ : Solve the second equation for $\tan(\theta)$ .
$\tan(\theta) = \frac{7}{2}$ Find the angles that correspond to $\tan(\theta) = -\frac{2}{3}$ in the range 0° \leq \theta < 360°.
Using a calculator, we find that $\theta \approx 180° - \arctan(\frac{2}{3})$ and $\theta \approx 360° - \arctan(\frac{2}{3})$ .Calculate the angles for $\tan(\theta) = -\frac{2}{3}$ .
$\theta \approx 180° - \arctan(\frac{2}{3}) \approx 180° - 33.7° \approx 146.3°$
$\theta \approx 360° - \arctan(\frac{2}{3}) \approx 360° - 33.7° \approx 326.3°$
Calculate Angles for $\frac{7}{2}$ : Solve the second equation for $\tan(\theta)$ . $\tan(\theta) = \frac{7}{2}$ Find the angles that correspond to $\tan(\theta) = -\frac{2}{3}$ in the range 0^\circ \leq \theta < 360^\circ. Using a calculator, we find that $\theta \approx 180^\circ - \arctan(\frac{2}{3})$ and $\theta \approx 360^\circ - \arctan(\frac{2}{3})$ .Calculate the angles for $\tan(\theta) = -\frac{2}{3}$ . $\theta \approx 180^\circ - \arctan(\frac{2}{3}) \approx 180^\circ - 33.7^\circ \approx 146.3^\circ$ $\theta \approx 360^\circ - \arctan(\frac{2}{3}) \approx 360^\circ - 33.7^\circ \approx 326.3^\circ$ Find the angles that correspond to $\tan(\theta) = \frac{7}{2}$ in the range 0^\circ \leq \theta < 360^\circ. Using a calculator, we find that $\tan(\theta)$ $2$ and $\tan(\theta)$ $3$ .
Calculate Angles for $\frac{7}{2}$ : Solve the second equation for $\tan(\theta)$ .
$\tan(\theta) = \frac{7}{2}$ Find the angles that correspond to $\tan(\theta) = -\frac{2}{3}$ in the range 0^\circ \leq \theta < 360^\circ.
Using a calculator, we find that $\theta \approx 180^\circ - \arctan(\frac{2}{3})$ and $\theta \approx 360^\circ - \arctan(\frac{2}{3})$ .Calculate the angles for $\tan(\theta) = -\frac{2}{3}$ .
$\theta \approx 180^\circ - \arctan(\frac{2}{3}) \approx 180^\circ - 33.7^\circ \approx 146.3^\circ$
$\theta \approx 360^\circ - \arctan(\frac{2}{3}) \approx 360^\circ - 33.7^\circ \approx 326.3^\circ$ Find the angles that correspond to $\tan(\theta) = \frac{7}{2}$ in the range 0^\circ \leq \theta < 360^\circ.
Using a calculator, we find that $\tan(\theta)$ $2$ and $\tan(\theta)$ $3$ .Calculate the angles for $\tan(\theta) = \frac{7}{2}$ .
$\tan(\theta)$ $5$
$\tan(\theta)$ $6$