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.3, tangent, theta, plus, 4, equals, 03tanθ+4=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.3, tangent, theta, plus, 4, equals, 03tanθ+4=0
Solution
To solve the equation 3tanθ + 4 = 0 for all values of θ such that 0 ≤ θ < 360, follow these steps:
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First, isolate tanθ in the equation. To do this, subtract 4 from both sides of the equation and then divide by 3:
3tanθ + 4 - 4 = 0 - 4 3tanθ = -4 tanθ = -4/3
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Next, find the value of θ by taking the arctangent (inverse tangent) of -4/3. Make sure your calculator is in degree mode:
θ = arctan(-4/3) ≈ -53.1 degrees
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Since the range of the arctangent function is -90 to 90 degrees, and we want 0 ≤ θ < 360, we need to add 180 degrees to the result to get a positive angle in the second quadrant:
θ = -53.1 + 180 = 126.9 degrees
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The tangent function has a period of 180 degrees, so to find the second solution, add 180 degrees to the first solution:
θ = 126.9 + 180 = 306.9 degrees
So, the solutions to the equation 3tanθ + 4 = 0 for 0 ≤ θ < 360 are θ = 126.9 degrees and θ = 306.9 degrees, rounded to the nearest tenth.
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