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Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. tan^(2)theta-3tan theta=0 Answer: theta=

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newlinetan⁡2θ−3tan⁡θ=0 \tan ^{2} \theta-3 \tan \theta=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.\newlinetan⁡2θ−3tan⁡θ=0 \tan ^{2} \theta-3 \tan \theta=0 \newlineAnswer: θ= \theta=
  1. Factor Quadratic Equation: Factor the given quadratic equation in terms of tan⁡(θ)\tan(\theta).tan⁡2(θ)−3tan⁡(θ)=0\tan^2(\theta) - 3\tan(\theta) = 0 can be factored as tan⁡(θ)(tan⁡(θ)−3)=0\tan(\theta)(\tan(\theta) - 3) = 0.
  2. Set Equations Equal: Set each factor equal to zero to find the values of θ\theta that satisfy the equation.tan⁡(θ)=0\tan(\theta) = 0 and tan⁡(θ)−3=0\tan(\theta) - 3 = 0.
  3. Solve tan⁡(θ)=0\tan(\theta) = 0: Solve the first equation tan⁡(θ)=0\tan(\theta) = 0.\newlineThe tangent function is zero at 00 degrees and 180180 degrees.\newlineθ=0\theta = 0 degrees, 180180 degrees.
  4. Solve tan⁡(θ)−3=0\tan(\theta) - 3 = 0: Solve the second equation tan⁡(θ)−3=0\tan(\theta) - 3 = 0.\newlinetan⁡(θ)=3\tan(\theta) = 3.\newlineUse an inverse tangent function to find the angle whose tangent is 33.\newlineθ≈arctan⁡(3)\theta \approx \arctan(3).
  5. Calculate arctan⁡(3)\arctan(3): Calculate the angle using a calculator for arctan⁡(3)\arctan(3).θ≈71.6\theta \approx 71.6 degrees.
  6. Find Second Solution: Find the second solution within the range 00 degrees to 360360 degrees.\newlineThe tangent function has a period of 180180 degrees, so we add 180180 degrees to the first solution.\newlineθ≈71.6\theta \approx 71.6 degrees +180+ 180 degrees =251.6= 251.6 degrees.
  7. List All Solutions: List all the solutions within the given range. θ=0\theta = 0 degrees, 71.671.6 degrees, 180180 degrees, 251.6251.6 degrees.

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