Solve for all values of thetaθ, such that 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , rounding all values to the nearest tenth.5, cosecant, theta, minus, 8, equals, 05cscθ−8=0
Question
Solve for all values of thetaθ, such that 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , rounding all values to the nearest tenth.5, cosecant, theta, minus, 8, equals, 05cscθ−8=0
Solution
To solve the equation 5cscθ - 8 = 0 for 0 ≤ θ < 360, follow these steps:
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First, isolate cscθ by adding 8 to both sides of the equation and then dividing by 5. This gives you cscθ = 8/5.
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The cosecant function is the reciprocal of the sine function. So, sinθ = 5/8.
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Now, we need to find the angles θ for which sinθ = 5/8.
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Use the inverse sine function to find the principal value: θ = arcsin(5/8) ≈ 38.7 degrees.
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The sine function is positive in both the first and second quadrants. So, the solutions in this interval are θ = 38.7 degrees and θ = 180 - 38.7 = 141.3 degrees.
So, the solutions to the equation 5cscθ - 8 = 0 for 0 ≤ θ < 360 are θ = 38.7 degrees and θ = 141.3 degrees, to the nearest tenth of a degree.
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