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Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. 9tan^(2)theta+19 tan theta=-2 Answer: theta=

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline9tan⁡2θ+19tan⁡θ=−2 9 \tan ^{2} \theta+19 \tan \theta=-2 \newlineAnswer: θ= \theta=

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Csc, sec, and cot of special anglesright chevron icon

Full solution

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.\newline9tan⁡2θ+19tan⁡θ=−2 9 \tan ^{2} \theta+19 \tan \theta=-2 \newlineAnswer: θ= \theta=
  1. Rewrite Equation in Quadratic Form: Rewrite the given equation in a quadratic form by moving all terms to one side.\newline9tan⁡2(θ)+19tan⁡(θ)+2=09\tan^2(\theta) + 19\tan(\theta) + 2 = 0
  2. Factor Quadratic Equation: Factor the quadratic equation to find the values of tan⁡(θ)\tan(\theta).(3tan⁡(θ)+1)(3tan⁡(θ)+2)=0 (3\tan(\theta) + 1)(3\tan(\theta) + 2) = 0
  3. Solve for tan⁡(θ)\tan(\theta): Set each factor equal to zero and solve for tan⁡(θ)\tan(\theta).\newlineFirst factor: 3tan⁡(θ)+1=03\tan(\theta) + 1 = 0\newlinetan⁡(θ)=−13\tan(\theta) = -\frac{1}{3}
  4. Find Corresponding Angles: Find the angles that correspond to the tangent values found in Step 33 within the range 0^\circ \leq \theta < 360^\circ. For tan⁡(θ)=−13\tan(\theta) = -\frac{1}{3}, use an inverse tangent function or a calculator to find the reference angle. θ=arctan⁡(−13)≈−18.4∘\theta = \arctan(-\frac{1}{3}) \approx -18.4^\circ Since the tangent function is negative in the second and fourth quadrants, add 180∘180^\circ to find the angle in the second quadrant and use the reference angle to find the angle in the fourth quadrant. Second quadrant: 180∘−18.4∘≈161.6∘180^\circ - 18.4^\circ \approx 161.6^\circ Fourth quadrant: 360∘−18.4∘≈341.6∘360^\circ - 18.4^\circ \approx 341.6^\circ
  5. List All Angles: List all the angles found in Step 44 that are within the range 0^\circ \leq \theta < 360^\circ.θ≈161.6∘,341.6∘,146.3∘,326.3∘\theta \approx 161.6^\circ, 341.6^\circ, 146.3^\circ, 326.3^\circ

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