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Find all solutions with −90∘≤θ≤90∘ -90^\circ \leq \theta \leq 90^\circ . Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas. −2sin⁡(θ)+92=−13sin⁡(θ)+10 - 2\sin(\theta)+ \frac{9}{2} = - 13\sin(\theta)+10

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Q. Find all solutions with −90∘≤θ≤90∘ -90^\circ \leq \theta \leq 90^\circ . Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas. −2sin⁡(θ)+92=−13sin⁡(θ)+10 - 2\sin(\theta)+ \frac{9}{2} = - 13\sin(\theta)+10
  1. Simplify Equation: Simplify the equation by moving all sin⁡(θ)\sin(\theta) terms to one side and constants to the other.\newline−2sin⁡(θ)+92=−13sin⁡(θ)+10-2\sin(\theta) + \frac{9}{2} = -13\sin(\theta) + 10\newlineAdd 13sin⁡(θ)13\sin(\theta) to both sides:\newline11sin⁡(θ)+92=1011\sin(\theta) + \frac{9}{2} = 10\newlineSubtract 92\frac{9}{2} from both sides:\newline11sin⁡(θ)=10−9211\sin(\theta) = 10 - \frac{9}{2}\newline11sin⁡(θ)=11211\sin(\theta) = \frac{11}{2}\newlineDivide both sides by 1111:\newlinesin⁡(θ)=112/11\sin(\theta) = \frac{11}{2} / 11\newlinesin⁡(θ)=12\sin(\theta) = \frac{1}{2}
  2. Add and Subtract: Find the values of θ\theta that satisfy sin⁡(θ)=12\sin(\theta) = \frac{1}{2} within the interval –90°≤θ≤90°–90°\leq\theta\leq90°.\newlinesin⁡(θ)=12\sin(\theta) = \frac{1}{2} at θ=30°\theta = 30° and θ=150°\theta = 150°, but since 150°150° is not within the interval –90°–90° to 90°90°, we only consider θ=30°\theta = 30°.

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