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.2, secant, theta, plus, 3, equals, 02secθ+3=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.2, secant, theta, plus, 3, equals, 02secθ+3=0
Solution
To solve the equation 2secθ + 3 = 0 for 0 ≤ θ < 360, follow these steps:
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First, isolate secθ by subtracting 3 from both sides of the equation and then dividing by 2. This gives us secθ = -3/2.
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The secant of an angle is the reciprocal of the cosine of the angle. So, we can rewrite the equation as cosθ = -2/3.
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Now, we need to find the angles θ for which the cosine is -2/3.
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Cosine is negative in the second and third quadrants. So, we find the reference angle in the first quadrant whose cosine is 2/3, and then find the angles in the second and third quadrants that have the same reference angle.
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The reference angle is cos^-1(2/3) ≈ 48.2 degrees.
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The angle in the second quadrant is 180 - 48.2 = 131.8 degrees.
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The angle in the third quadrant is 180 + 48.2 = 228.2 degrees.
So, the solutions to the equation are θ = 131.8 degrees and θ = 228.2 degrees, to the nearest tenth of a degree.
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