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Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. 9cot^(2)theta-1=0 Answer: theta=

Find all angles, 0∘≤θ<360∘ 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline9cot⁡2θ−1=0 9 \cot ^{2} \theta-1=0 \newlineAnswer: θ= \theta=

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Csc, sec, and cot of special anglesright chevron icon

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Q. Find all angles, 0∘≤θ<360∘ 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline9cot⁡2θ−1=0 9 \cot ^{2} \theta-1=0 \newlineAnswer: θ= \theta=
  1. Solve for cotangent: Solve the equation for cot⁡2(θ)\cot^2(\theta). \newline9cot⁡2(θ)−1=09\cot^2(\theta) - 1 = 0\newlineAdd 11 to both sides of the equation.\newline9cot⁡2(θ)=19\cot^2(\theta) = 1\newlineDivide both sides by 99.\newlinecot⁡2(θ)=19\cot^2(\theta) = \frac{1}{9}\newlineTake the square root of both sides.\newlinecot⁡(θ)=±13\cot(\theta) = \pm\frac{1}{3}
  2. Find corresponding angles: Find the angles that correspond to the cotangent values.\newlineSince cotangent is the reciprocal of tangent, we can write:\newlinetan⁡(θ)=±3\tan(\theta) = \pm 3
  3. Positive tangent value: Determine the angles for the positive tangent value.\newlineUsing a calculator or trigonometric tables, find the angle whose tangent is 33.\newlinetan⁡−1(3)≈71.6∘\tan^{-1}(3) \approx 71.6^\circ\newlineSince tangent is positive in the first and third quadrants, we add 180∘180^\circ to find the angle in the third quadrant.\newline71.6∘+180∘=251.6∘71.6^\circ + 180^\circ = 251.6^\circ
  4. Negative tangent value: Determine the angles for the negative tangent value.\newlineSince tangent is negative in the second and fourth quadrants, we find the reference angle for tan⁡−1(−3)\tan^{-1}(-3).\newlinetan⁡−1(−3)≈−71.6∘\tan^{-1}(-3) \approx -71.6^\circ\newlineTo find the angle in the second quadrant, we add 180∘180^\circ to the reference angle.\newline180∘−71.6∘=108.4∘180^\circ - 71.6^\circ = 108.4^\circ\newlineTo find the angle in the fourth quadrant, we add 360∘360^\circ to the reference angle.\newline360∘−71.6∘=288.4∘360^\circ - 71.6^\circ = 288.4^\circ
  5. List of satisfying angles: List all the angles that satisfy the original equation. θ≈71.6∘,108.4∘,251.6∘,288.4∘\theta \approx 71.6^\circ, 108.4^\circ, 251.6^\circ, 288.4^\circ

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