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Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. sin^(2)theta+sin 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.\newlinesin⁡2θ+sin⁡θ=0 \sin ^{2} \theta+\sin \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.\newlinesin⁡2θ+sin⁡θ=0 \sin ^{2} \theta+\sin \theta=0 \newlineAnswer: θ= \theta=
  1. Factor Trigonometric Equation: Factor the given trigonometric equation.\newlineWe are given sin⁡2(θ)+sin⁡(θ)=0\sin^2(\theta) + \sin(\theta) = 0. We can factor out sin⁡(θ)\sin(\theta) from both terms.\newlinesin⁡(θ)⋅(sin⁡(θ)+1)=0\sin(\theta) \cdot (\sin(\theta) + 1) = 0
  2. Set Factors Equal to Zero: Set each factor equal to zero to find the solutions for θ\theta.sin⁡(θ)=0\sin(\theta) = 0 and sin⁡(θ)+1=0\sin(\theta) + 1 = 0
  3. Solve for sin⁡(θ)=0\sin(\theta) = 0: Solve the first equation sin⁡(θ)=0\sin(\theta) = 0. The sine of an angle is zero at 00 degrees, 180180 degrees, and 360360 degrees. Therefore, θ=0\theta = 0 degrees, 180180 degrees, and 360360 degrees.
  4. Solve for sin⁡(θ)=−1\sin(\theta) = -1: Solve the second equation sin⁡(θ)+1=0\sin(\theta) + 1 = 0.\newlineThis implies sin⁡(θ)=−1\sin(\theta) = -1.\newlineThe sine of an angle is −1-1 at 270270 degrees.\newlineTherefore, θ=270\theta = 270 degrees.
  5. Combine All Solutions: Combine all the solutions.\newlineThe angles that satisfy the given equation are 00 degrees, 180180 degrees, 270270 degrees, and 360360 degrees.

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