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Find all angles,
0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree.

cot^(2)theta+4cot theta+3=0
Answer:
theta=

Find all angles, $0^{\circ} \leq \theta<360^{\circ}$ , that satisfy the equation below, to the nearest tenth of a degree. $\newline$ $\cot ^{2} \theta+4 \cot \theta+3=0$ $\newline$ Answer: $\theta=$

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Find the roots of factored polynomials

Full solution

Q. Find all angles,

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

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

\newline

\cot ^{2} \theta+4 \cot \theta+3=0

\newline

Answer:

\theta=

Solve Quadratic Equation: Let's first solve the quadratic equation in terms of $\cot(\theta)$ . The equation is $\cot^2(\theta) + 4\cot(\theta) + 3 = 0$ . This is a quadratic equation in standard form, where $a = 1$ , $b = 4$ , and $c = 3$ .
Factor Quadratic Equation: We can factor the quadratic equation as $(\cot(\theta) + 1)(\cot(\theta) + 3) = 0$ . This gives us two possible solutions for $\cot(\theta)$ : $\cot(\theta) = -1$ and $\cot(\theta) = -3$ .
Find Theta for Cot( $-1$ ): Now we need to find the angles $\theta$ that correspond to these cotangent values. For $\cot(\theta) = -1$ , we know that cotangent is the reciprocal of tangent, so we are looking for angles where $\tan(\theta) = -1$ . The angles with tangent of $-1$ in the range of $0$ to $360$ degrees are $135$ degrees and $315$ degrees.
Find Theta for Cot( $-3$ ): For $\cot(\theta) = -3$ , we need to use a calculator to find the angles that correspond to this cotangent value. Since cotangent is the reciprocal of tangent, we are looking for angles where $\tan(\theta) = -\frac{1}{3}$ . We can use the inverse tangent function to find the reference angle, and then determine the angles in the specified range that have this tangent value.
Combine Results: Using a calculator, we find that the reference angle for $\tan(\theta) = -\frac{1}{3}$ is approximately $18.4$ degrees. Since the tangent is negative in the second and fourth quadrants, the angles that satisfy this condition are $180 - 18.4 = 161.6$ degrees and $360 - 18.4 = 341.6$ degrees.
Combine Results: Using a calculator, we find that the reference angle for $\tan(\theta) = -\frac{1}{3}$ is approximately $18.4$ degrees. Since the tangent is negative in the second and fourth quadrants, the angles that satisfy this condition are $180 - 18.4 = 161.6$ degrees and $360 - 18.4 = 341.6$ degrees.Combining the results from the two factors, we have four angles that satisfy the original equation: $135$ degrees, $315$ degrees, $161.6$ degrees, and $341.6$ degrees.

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