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Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. 8cos^(2)theta+6cos theta-2=-3

Find all angles, 0∘≤θ<360∘ 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline8cos⁡2θ+6cos⁡θ−2=−3 8 \cos ^{2} \theta+6 \cos \theta-2=-3

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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.\newline8cos⁡2θ+6cos⁡θ−2=−3 8 \cos ^{2} \theta+6 \cos \theta-2=-3
  1. Rewrite Equation: Rewrite the equation to have zero on one side.\newline8cos⁡2(θ)+6cos⁡(θ)−2+3=08\cos^2(\theta) + 6\cos(\theta) - 2 + 3 = 0\newline8cos⁡2(θ)+6cos⁡(θ)+1=08\cos^2(\theta) + 6\cos(\theta) + 1 = 0
  2. Factor Quadratic: Factor the quadratic equation.\newline(2cos⁡(θ)+1)(4cos⁡(θ)+1)=0(2\cos(\theta) + 1)(4\cos(\theta) + 1) = 0
  3. Solve for cos⁡(θ)\cos(\theta): Solve for cos⁡(θ)\cos(\theta) by setting each factor equal to zero.2cos⁡(θ)+1=02\cos(\theta) + 1 = 0 and 4cos⁡(θ)+1=04\cos(\theta) + 1 = 0
  4. Find Angles: Solve the first equation for cos⁡(θ)\cos(\theta).cos⁡(θ)=−12\cos(\theta) = -\frac{1}{2}
  5. Check Range: Find the angles for cos⁡(θ)=−12\cos(\theta) = -\frac{1}{2}.\newlineθ=120\theta = 120 degrees and θ=240\theta = 240 degrees
  6. Check Range: Find the angles for cos⁡(θ)=−12\cos(\theta) = -\frac{1}{2}.θ=120\theta = 120 degrees and θ=240\theta = 240 degrees Solve the second equation for cos⁡(θ)\cos(\theta).4cos⁡(θ)+1=04\cos(\theta) + 1 = 0cos⁡(θ)=−14\cos(\theta) = -\frac{1}{4}
  7. Check Range: Find the angles for cos⁡(θ)=−12\cos(\theta) = -\frac{1}{2}.θ=120\theta = 120 degrees and θ=240\theta = 240 degrees Solve the second equation for cos⁡(θ)\cos(\theta).4cos⁡(θ)+1=04\cos(\theta) + 1 = 0cos⁡(θ)=−14\cos(\theta) = -\frac{1}{4}Find the angles for cos⁡(θ)=−14\cos(\theta) = -\frac{1}{4} using a calculator.θ≈104.5\theta \approx 104.5 degrees and θ≈255.5\theta \approx 255.5 degrees
  8. Check Range: Find the angles for cos⁡(θ)=−12\cos(\theta) = -\frac{1}{2}.θ=120\theta = 120 degrees and θ=240\theta = 240 degrees Solve the second equation for cos⁡(θ)\cos(\theta).4cos⁡(θ)+1=04\cos(\theta) + 1 = 0cos⁡(θ)=−14\cos(\theta) = -\frac{1}{4}Find the angles for cos⁡(θ)=−14\cos(\theta) = -\frac{1}{4} using a calculator.θ≈104.5\theta \approx 104.5 degrees and θ≈255.5\theta \approx 255.5 degrees Check if all found angles are within the specified range.0≤θ<3600 \leq \theta < 360All angles are within the range.

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