Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree. $\newline$ $9 \cot ^{2} \theta-1=0$ $\newline$ 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

9 \cot ^{2} \theta-1=0

\newline

Answer:

\theta=

Solve for cotangent: Solve the equation for $\cot^2(\theta)$ . $\newline$ $9\cot^2(\theta) - 1 = 0$ $\newline$ Add $1$ to both sides of the equation. $\newline$ $9\cot^2(\theta) = 1$ $\newline$ Divide both sides by $9$ . $\newline$ $\cot^2(\theta) = \frac{1}{9}$ $\newline$ Take the square root of both sides. $\newline$ $\cot(\theta) = \pm\frac{1}{3}$
Find corresponding angles: Find the angles that correspond to the cotangent values. $\newline$ Since cotangent is the reciprocal of tangent, we can write: $\newline$ $\tan(\theta) = \pm 3$
Positive tangent value: Determine the angles for the positive tangent value. $\newline$ Using a calculator or trigonometric tables, find the angle whose tangent is $3$ . $\newline$ $\tan^{-1}(3) \approx 71.6^\circ$ $\newline$ Since tangent is positive in the first and third quadrants, we add $180^\circ$ to find the angle in the third quadrant. $\newline$ $71.6^\circ + 180^\circ = 251.6^\circ$
Negative tangent value: Determine the angles for the negative tangent value. $\newline$ Since tangent is negative in the second and fourth quadrants, we find the reference angle for $\tan^{-1}(-3)$ . $\newline$ $\tan^{-1}(-3) \approx -71.6^\circ$ $\newline$ To find the angle in the second quadrant, we add $180^\circ$ to the reference angle. $\newline$ $180^\circ - 71.6^\circ = 108.4^\circ$ $\newline$ To find the angle in the fourth quadrant, we add $360^\circ$ to the reference angle. $\newline$ $360^\circ - 71.6^\circ = 288.4^\circ$
List of satisfying angles: List all the angles that satisfy the original equation. $\theta \approx 71.6^\circ, 108.4^\circ, 251.6^\circ, 288.4^\circ$