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Find all solutions with −π2≤θ≤π2 - \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} . Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas. 14csc⁡(θ)+14=014\csc(\theta)+14=0

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Q. Find all solutions with −π2≤θ≤π2 - \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} . Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas. 14csc⁡(θ)+14=014\csc(\theta)+14=0
  1. Simplify equation: Step 11: Simplify the equation 14csc⁡(θ)+14=014\csc(\theta) + 14 = 0. We start by isolating csc⁡(θ)\csc(\theta): 14csc⁡(θ)=−1414\csc(\theta) = -14 csc⁡(θ)=−1414\csc(\theta) = -\frac{14}{14} csc⁡(θ)=−1\csc(\theta) = -1
  2. Convert to sin: Step 22: Convert csc(θ)csc(\theta) to sin(θ)sin(\theta).\newlineSince csc(θ)=1sin(θ)csc(\theta) = \frac{1}{sin(\theta)}, we have:\newline1sin(θ)=−1\frac{1}{sin(\theta)} = -1\newlinesin(θ)=−1sin(\theta) = -1
  3. Find solution: Step 33: Find θ\theta such that sin⁡(θ)=−1\sin(\theta) = -1 within the interval −π/2≤θ≤π/2-\pi/2 \leq \theta \leq \pi/2.\newlineThe sine function equals −1-1 at θ=−π/2\theta = -\pi/2.\newlineThus, θ=−π/2\theta = -\pi/2 is the solution in the given interval.

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