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Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. 16cot^(2)theta-1=0 Answer: theta=

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline16cot⁡2θ−1=0 16 \cot ^{2} \theta-1=0 \newlineAnswer: θ= \theta=

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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.\newline16cot⁡2θ−1=0 16 \cot ^{2} \theta-1=0 \newlineAnswer: θ= \theta=
  1. Solve for cot⁡2(θ)\cot^2(\theta): Solve the equation for cot⁡2(θ)\cot^2(\theta).
    16cot⁡2(θ)−1=016\cot^2(\theta) - 1 = 0
    Add 11 to both sides of the equation.
    16cot⁡2(θ)=116\cot^2(\theta) = 1
    Divide both sides by 1616.
    cot⁡2(θ)=116\cot^2(\theta) = \frac{1}{16}
    Take the square root of both sides.
    cot⁡(θ)=±14\cot(\theta) = \pm\frac{1}{4}
  2. Find positive cotangent angles: Find the angles that correspond to the positive cotangent value.\newlinecot⁡(θ)=14\cot(\theta) = \frac{1}{4}\newlineSince cotangent is the reciprocal of tangent, we have:\newlinetan⁡(θ)=4\tan(\theta) = 4\newlineUse the arctangent function to find the angle whose tangent is 44.\newlineθ=arctan⁡(4)\theta = \arctan(4)\newlineUse a calculator to find the value of θ\theta.\newlineθ≈75.96\theta \approx 75.96 degrees\newlineHowever, cotangent is positive in the first and third quadrants, so we need to find the angle in the third quadrant as well.\newlineθ=180+75.96\theta = 180 + 75.96\newlineθ≈255.96\theta \approx 255.96 degrees
  3. Find negative cotangent angles: Find the angles that correspond to the negative cotangent value.\newlinecot⁡(θ)=−14\cot(\theta) = -\frac{1}{4}\newlineSince cotangent is the reciprocal of tangent, we have:\newlinetan⁡(θ)=−4\tan(\theta) = -4\newlineUse the arctangent function to find the angle whose tangent is −4-4.\newlineθ=arctan⁡(−4)\theta = \arctan(-4)\newlineUse a calculator to find the value of θ\theta.\newlineθ≈−75.96\theta \approx -75.96 degrees\newlineHowever, we need angles between 00 and 360360 degrees, so we add 180180 degrees to find the angle in the second quadrant.\newlineθ=180−75.96\theta = 180 - 75.96\newlineθ≈104.04\theta \approx 104.04 degrees\newlineAnd add 360360 degrees to find the angle in the fourth quadrant.\newlinetan⁡(θ)=−4\tan(\theta) = -411\newlinetan⁡(θ)=−4\tan(\theta) = -422 degrees
  4. List all satisfying angles: List all the angles that satisfy the equation.\newlineWe have found four angles that satisfy the equation:\newlineθ≈75.96\theta \approx 75.96 degrees (first quadrant)\newlineθ≈255.96\theta \approx 255.96 degrees (third quadrant)\newlineθ≈104.04\theta \approx 104.04 degrees (second quadrant)\newlineθ≈284.04\theta \approx 284.04 degrees (fourth quadrant)\newlineRound each angle to the nearest tenth of a degree.\newlineθ≈76.0\theta \approx 76.0 degrees\newlineθ≈256.0\theta \approx 256.0 degrees\newlineθ≈104.0\theta \approx 104.0 degrees\newlineθ≈284.0\theta \approx 284.0 degrees

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